Title Study on Upward Turbulent Bubbly Flow in Ducts ...repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/...express how grateful I am to my father Mr. Liu-Xian ZHANG, my mother

Title Study on Upward Turbulent Bubbly Flow in Ducts(Dissertation_全文 )

Author(s) Zhang, Hongna

Citation Kyoto University (京都大学)

Issue Date 2014-09-24

URL https://doi.org/10.14989/doctor.k18590

Right 許諾条件により要旨は2014/12/24に公開

Type Thesis or Dissertation

Textversion ETD

Kyoto University

Study on Upward Turbulent Bubbly Flow in Ducts

ZHANG HONGNA

Doctoral Dissertation

Department of Nuclear Engineering

Graduate School of Engineering

Kyoto University

September 2014

i

Acknowledgements

I here would like to sincerely thank many people for their kind aids and

encouragements during my doctor course in Kyoto University.

Foremost, I would like to express my special appreciation and thanks to my

supervisor Profess Tomaki KUNUGI, who has been a tremendous mentor of me for his

patience, motivation, enthusiasm, and immense knowledge. I would like to thank him

for showing me what the research is and how to enjoy it, for discussing on many

scientific problems, for encouraging me, and for allowing me to grow as a research

scientist freely. I have been extremely lucky to have a supervisor like him.

Special thanks are also extended to my committee members, Professor

Kazuyoshi NAKABE and Associate professor Takehiko YOKOMINE for serving as

my committee members and for their questions and suggestions during my defense.

I also wish to thank all the members in KUNUGI Lab for their help on my life

and research in Kyoto University. Special thanks are given to Professor Jun

SUGIMOTO, Lecturer Zensaku KAWARA, Dr. Li-Fang JIAO, Dr. Hao-Min SUN, Dr.

Hong-Son PHAM and Dr. Yang CAO for their help on my life and their discussions on

my research and Ms. Hijiri SOEJIMA for her management on my financial budget.

Thanks are also extended to Japan Society for the Promotion of Science (JSPS)

Research Fellowship and Global Center of Excellence (G-COE) Program (J-051) for

their financial support during my doctor course.

Finally, special thanks are given to my family and my friends. Words cannot

express how grateful I am to my father Mr. Liu-Xian ZHANG, my mother Mrs.

Feng-Ying LI, and my brother Mr. Chao ZHANG for all their sacrifices,

encouragements and supports that they’ve made on my behalf. I would also like to

thank all of my friends who supported me in writing, and encouraged me to strive

towards my goal.

ii

iii

Table of Contents

Acknowledgements .......................................................................................................... i

List of Tables ................................................................................................................. vii

List of Figures .............................................................................................................. viii

Chapter 1 Introduction ............................................................................................... 1

1.1 Motivation .......................................................................................................... 1

1.2 State-of-the-art of turbulent bubbly flow............................................................ 2

1.2.1 Background .................................................................................................. 2

1.2.2 Review of the experimental and numerical databases ................................. 5

1.2.3 Review of existing models for turbulent bubbly flow ............................... 17

1.2.4 Summary on review of databases and models ........................................... 28

1.3 Scope and layout of the dissertation ................................................................. 29

Reference .................................................................................................................... 30

Chapter 2 Experimental databases of upward turbulent bubbly flow ................. 37

2.1 Introduction ...................................................................................................... 37

2.2 Experimental techniques................................................................................... 37

2.3 Experimental apparatus .................................................................................... 38

2.4 Experimental results ......................................................................................... 42

2.5 Summary ........................................................................................................... 43

Reference .................................................................................................................... 51

Chapter 3 Numerical evaluation of two-fluid model of turbulent bubbly flow in

large square duct .......................................................................................................... 53

3.1 Introduction ...................................................................................................... 53

3.2 Numerical methodology ................................................................................... 53

3.2.1 Problem description ................................................................................... 53

3.2.2 Numerical methods .................................................................................... 54

iv

3.3 Results and discussion ...................................................................................... 59

3.4 Summary ........................................................................................................... 60

Reference .................................................................................................................... 63

Chapter 4 Developing process of turbulent bubbly flow ....................................... 65

4.1 Introduction ...................................................................................................... 65

4.2 Experimental observations ............................................................................... 65

4.3 Modeling of developing process ...................................................................... 71

4.3.1 Formation of bubble layer in the first stage ............................................... 71

4.3.2 Liquid phase velocity lifting in the second stage ...................................... 74

4.4 Summary and Discussion ................................................................................. 77

Reference .................................................................................................................... 78

Chapter 5 Axial liquid-phase velocity of turbulent bubbly flow ........................... 79

5.1 Introduction ...................................................................................................... 79

5.2 Index of flow type .............................................................................................. 80

5.3 Conditions for M-shaped Wl profile ................................................................... 89

5.4 Duct geometry effect ........................................................................................ 90

5.5 Peak location of Wl near the wall ..................................................................... 95

5.6 Flow characteristics in two layers .................................................................... 95

5.6.1 Definition of two layers ............................................................................. 96

5.6.2 Velocity characteristics .............................................................................. 97

5.7 Numerical validation of the liquid phase velocity .......................................... 101

5.7.1 Numerical models .................................................................................... 101

5.7.2 Numerical results ..................................................................................... 103

5.8 Summary and discussion ................................................................................ 105

Reference .................................................................................................................. 107

Chapter 6 Void fraction of the turbulent bubbly flow ......................................... 109

v

6.1 Introduction .................................................................................................... 109

6.2 Bubble deformation ......................................................................................... 110

6.2.1 Definition of bubbles’ statistical shape ..................................................... 110

6.2.2 Distribution of bubbles’ shape factor ........................................................ 117

6.2.3 Void fraction effects on shape factor ....................................................... 121

6.2.4 Bubble size validation ............................................................................. 127

6.3 Numerical validation of the void fraction....................................................... 129

6.3.1 Numerical models .................................................................................... 129

6.3.2 Numerical results ..................................................................................... 131

6.4 Summary and Discussion ............................................................................... 136

Reference .................................................................................................................. 137

Chapter 7 Turbulence modulation of the turbulent bubbly flow ........................ 139

7.1 Introduction .................................................................................................... 139

7.2 Turbulent kinetic energy ................................................................................. 140

7.2.1 Bubble-induced turbulence at the duct center ......................................... 147

7.2.2 Bubble-induced and wall-induced turbulence interaction ....................... 154

7.3 Turbulence anisotropy in turbulent bubbly flow ............................................ 166

7.3.1 Bubble effects on turbulence anisotropy in liquid phase ......................... 166

7.3.2 Modeling of the anisotropic turbulent bubbly flow ................................. 174

7.4 Summary and discussion ................................................................................ 180

Reference .................................................................................................................. 182

Chapter 8 Conclusions ............................................................................................ 185

8.1 Summary of dissertation ................................................................................. 185

8.2 Recommendation for future work................................................................... 187

Nomenclature .............................................................................................................. 189

vi

vii

List of Tables

Table 1.1 Numerical databases based on DNSs of the turbulent bubbly flow.

Table 2.1 Experimental databases in different duct geometries. B: Base; H: Height.

Table 2.2 Experimental flow conditions in different duct geometries. B: Bubbly flow;

T: Transition regime.

Table 4.1 Experimental conditions for the developing process.

Table 5.1 Experimental data of dp/dz (kPa/m).

Table 5.2 Geometry dimension H to determine the wall shear stress

Table 5.3 Experimental conditions to for different geometries in the literatures. In the

table, C: circular duct; T: triangle duct; S: square duct; B: base length; H:

height.

viii

List of Figures

Fig. 2.1 (a) Experimental techniques for the flow measurements and (b) its

applications in the turbulent bubbly flow.

Fig. 2.2 Simplified schematic diagram of the experimental apparatus.

Fig. 2.3 Lateral distributions of axial liquid phase velocities Wl and void fraction α

in the small circular pipe with DH=60mm under (a) Jl =0.74m/s and (b) Jl

=1.03m/s based on Serizawa’s experimental database [2.1].

Fig. 2.4 Lateral distributions of axial turbulent intensities <ww> in the small circular

pipe with DH=60mm based on Serizawa’s experimental database: under (a)

Jl =0.74m/s and (b) Jl =1.03m/s [2.1].


in the small circular pipe with DH=50.8mm based on Hibiki’s experimental

database: under (a) Jl =0.491m/s and (b) Jl =0.986m/s [2.6].


pipe with DH=50.8mm based on Hibiki’s experimental database: under (a)

Jl =0.491m/s and (b) Jl =0.986m/s [2.6].

Fig. 2.7 Lateral distributions of axial turbulent kinetic energy <ww> and lateral

turbulent kinetic energy <uu> in the small circular pipe in the small circular

pipe with DH=60mm based on Liu’s experimental database [2.3-2.4].


in the large circular pipe with DH=200mm based on Shakwat’s experimental



turbulent kinetic energy <uu> in the large circular pipe with DH=200mm

based on Shakwat’s experimental database: under (a) Jl =0.45m/s and (b) Jl

=0.68m/s [2.7].


ix

in the large circular pipe with DH=136mm based on Sun’s experimental

database: under Jl =0.75m/s (a) along the bisector line; (b) along the

diagonal line; under Jl=1.00m/s (c) along the bisector line; (d) along the

diagonal line [2.10].


turbulent kinetic energy <uu> in the large circular pipe with DH=136mm

based on Sun’s experimental database: under Jl =0.75m/s (a) along the

bisector line; (b) along the diagonal line; under Jl=1.00m/s (c) along the

bisector line; (d) along the diagonal line [2.10].

Fig. 3.1 Schematic diagram of the numerical problem.

Fig. 3.2 Flow charts of numerical calculation.

Fig. 3.3 Comparisons of the experimental measurements and numericla predictions

under Jl =0.75m/s and Jg =0.09m/s: for the liquid phase velocity (a) along

the bisector line and (b) along diagonal line; for the void fraction (c) along

the bisector line and (d) along diagonal line; for turbulence (e) along the

bisector line and (f) along diagonal line;

Fig. 4.1 Liquid phase velocities Wl and the corresponding void fractions α at z=4.5

DH, 10 DH and 16DH for Jl=0.5m/s: with Jg=0.09m/s along (a) the bisector

line; (b) the diagonal line; with Jg=0.135m/s along (c) the bisector line; (d)

the diagonal line [4.7].



line; (b) the diagonal line; with Jg=0.135m/s along (c)the bisector line; (d)

the diagonal line; with Jg=0.18m/s along (e) the bisector line; (f) the




x

line; (b) the diagonal line; with Jg=0.135m/s along (c) the bisector line; (d)

the diagonal line; with Jg=0.18m/s along (e) the bisector line; (f) the



DH, 10 DH and 16DH for Jl=1.25m/s: with Jg=0.135m/s along (a) the

bisector line; (b) the diagonal line; with Jg=0.18m/s along (c) the bisector

line; (d) the diagonal line; with Jg=0.225m/s along (e) the bisector line; (f)

the diagonal line [4.7].

Fig. 4.5 Schematic diagram of the developing process of the upward turbulent

bubbly flow in the large square duct.

Fig. 4.6 Schematic diagram of the formation of the bubble layer due to the lift force

exerted on the bubbles and the developing of the boundary layer in the

liquid phase.

Fig. 4.7 Comparison between the thickness of the boundary layer δl and the bubble

layer δB at the measurement location z=4.5DH.

Fig. 4.8 Schematic diagram of liquid phase velocity lifting up in the second stage.

Fig. 4.9 Comparison of dimensionless parameter 2/ 1pA L g W for

different superficial liquid and gas phase velocities Jl and Jg and

measurement locations.

Fig. 5.1 Schematic diagram of the type of the velocity profile of the upward flow:

(a) M-shape; (b) ∩-shape; (c) Cuspid-shape; (d) W-shape

Fig. 5.2 Distributions of the local net driving force F for (a) Jl=0.75m/s and (c)

Jl=1.0m/s and corresponding liquid velocities Wl for (b) Jl=0.75m/s and (d)

Jl=1.0 m/s along the bisector line.

Fig. 5.3 Distributions of the local net driving force F for (a) Jl=0.75m/s and (c)

Jl=1.0m/s and corresponding liquid velocities Wl for (b) Jl=0.75m/s and (d)

Jl=1.0m/s along the diagonal line.

xi

Fig. 5.4 Schematic diagram of flow image for the small void fraction.

Fig. 5.5 Schematic diagram of flow image for the large void fraction.

Fig. 5.6 Schematic diagram of velocity peak location.

Fig. 5.7 Comparison of γ for different duct geometries for Jl around (a) 0.75m/s and

(b) 1.0m/s. For noncircular geometry, ―-1‖ and ―-2‖ indicate data obtained

near the wall center and corner, respectively.

Fig. 5.8 Definition of two layers

Fig. 5.9 Velocity characteristics in the inner layer along (a) the bisector line and (b)

diagonal line; and in the outer layer (c) along the bisector line and (d) the

diagonal line. The mean void fraction could be approximated as

αgm=Jg/(Jg+Jl).

Fig. 5.10 Estimation of lB-B from the experiments. T, TB and TL are total measuring

time, bubbles passing time and liquid passing time, respetively. WB is the

mean bubble velocity.

Fig. 5.11 Comparison between numerical predicted liquid phase velocity Wl and

experimental measurement on the bisector line for (a) Jl =0.75m/s and (b) Jl

=1.0m/s, respectively.

Fig. 5.12 Lateral distribution of mean viscous force ηv based on the experimental

measurements for (a) Jl =0.75m/s and (b) Jl =1.0m/s, respectively.

Fig. 6.1 The distributions of the Sauter mean diameter DSM and the chord length Lc

along the bisector line under different flow conditions :(a) Jl = 0.5m/s; (b) Jl

= 0.75m/s; (c) Jl = 1.0m/s; (d) Jl = 1.25m/s. The solid and dashed lines show

DSM, and Lc respectively.


along the diagonal line under different flow conditions :(a) Jl = 0.5m/s; (b)

Jl = 0.75m/s; (c) Jl = 1.0m/s; (d) Jl = 1.25m/s. The solid and dashed lines

show DSM, and Lc respectively.

xii


along the near-wall line under different flow conditions :(a) Jl = 0.5m/s; (b)

Jl = 0.75m/s; (c) Jl = 1.0m/s; (d) Jl = 1.25m/s. The solid and dashed lines

show DSM, and Lc respectively.

Fig. 6.4 Schematic diagram of the ellipsoidal bubble shape.

Fig. 6.5 The aspect ratio E vs. the ratio DSM/Lc for the ellipsoidal bubbles.

Fig. 6.6 The distributions of ratio r along bisector line under different flow

conditions: (a) Jl = 0.5m/s; (b) Jl = 0.75m/s; (c) Jl = 1.0m/s; (d) Jl = 1.25m/s.

Fig. 6.7 The distributions of the ratio r along the diagonal line under different flow


Fig. 6.8 The distributions of the ratio r along the near-wall line under different flow


Fig. 6.9 The distributions of the ratio E/E0 along the bisector line under different

flow conditions: (a) Jl = 0.5m/s; (b) Jl = 0.75m/s; (c) Jl = 1.0m/s; (d) Jl =

1.25m/s.

Fig. 6.10 The distributions of the ratio E/E0 along the diagonal line under different


1.25m/s.

Fig. 6.11 The distributions of the ratio E/E0 along the near-wall line under different


1.25m/s.

Fig. 6.12 The ratio R=Eα/ES vs. the liquid fraction (1-α) under all the tested flow

conditions.

Fig. 6.13 DH.exp vs. DHS under all the tested flow conditions.

Fig. 6.14 DH.exp vs. DHα under all the tested flow conditions.

Fig. 6.15 Schematic diagram of the turbulent components along the diagonal line.

Fig. 6.16 Comparison between numerical predicted void fraction and experimental

xiii

measurement for Jl =0.5m/s at z=16DH.





Fig. 7.1 Lateral distributions of the ratio of the turbulent kinetic energies between tkt

and tks in the small circular pipe with DH=60mm based on Serizawa’s

experimental database: under (a) Jl =0.74m/s and (b) Jl =1.03m/s [7.2].


and tks in the small circular pipe with DH=50.8mm based on Hibiki’s



and tks in the small circular pipe with DH=60mm based on Liu’s

experimental database [7.7].


and tks in the large circular pipe with DH=200mm based on Shakwat’s



and tks in the large circular pipe with DH=136mm based on Sun’s

experimental database: under (a) Jl =0.75m/s along the bisector line; (b) Jl

=0.75m/s along the diagonal line; (c) Jl =1.0m/s along the bisector line; (b)

Jl =1.0m/s along the diagonal line [7.16].

Fig. 7.6 (a) The relation between the ratio tkt / tks and the local void fraction α in the

duct center for different ducts; (b) The critical flow conditions Jl and Jg for

tkt / tks ≈ 25.

Fig. 7.7 (a) The axial turbulent intensities <ww> vs. the local void fraction α in the

duct center; (b) <ww>/Wl vs. the local void fraction α in the duct center.

xiv

Fig. 7.8 Comparison the experimental data of the turbulent kinetic energy tk of the

liquid phase with the new correlation Eq. (7.7) for different bubble diameter

at the duct center.

Fig. 7.9 Lateral distributions of the ratio of the local turbulent kinetic energy tkt to

that in the duct center c

ktt

in the small circular pipe with DH=60mm based

on Serizawa’s experimental database: under (a) Jl =0.74m/s and (b) Jl

=1.03m/s based on Serizawa’s experimental database [7.2].



ktt

in the small circular pipe with DH=50.8mm

based on Hibiki’s experimental database: under (a) Jl =0.491m/s and (b) Jl

=0.986m/s [7.15].



ktt

in the small circular pipe with DH=60mm based

on Liu’s experimental database [7.7].


that in the duct center

c

ktt in the large circular pipe with DH=200mm based

on Shakwat’s experimental database: under (a) Jl =0.45m/s and (b) Jl

=0.68m/s [7.13].


that in the duct center

c

ktt in the large noncircular pipe with DH=136mm

based on Sun’s experimental database: under Jl =0.75m/s (a) along the

bisector line; (b) along the diagonal line; under Jl =1.0m/s (c) along the

bisector line; (b) along the diagonal line [7.16].

Fig. 7.14 Comparison between , , ,/B w B w B c

kw kb kbt t t and /w c based on Sun’s


Fig. 7.15 (a) Relation between /w c and c ; (b) Relation between

, , ,/B w B w B c

kw kb kbt t t and c based on Sun’s experimental database [7.16].

xv

Fig. 7.16 Schematic diagram of the relation between the turbulence in the liquid

phase and the void fraction for the turbulent bubbly flow.

Fig. 7.17 Experimental conditions from [7.8] with turbulence suppression and

enhancement in the liquid phase.

Fig. 7.18 Comparison the experimental measured void fraction and the calculated

void fraction based on 2, /2 R

wS

kwcalc Wt near the wall.

Fig. 7.19 Schematic diagram of the mechanism for the turbulence suppression

Fig. 7.20 Lateral distributions of the turbulence anisotropy tensor b11 and b33 in the

small circular pipe with DH=60mm based on Serizawa’s experimental



small circular pipe with DH=57.2mm based on Liu’s experimental database

[7.7].


small circular pipe with DH=57.15mm based on Wang’s experimental

database [7.6].


large circular pipe with DH=200mm based on Shakwat’s experimental



large circular pipe with DH=136mm based on Sun’s experimental database:

under (a) Jl =0.75m/s along the bisector line; (b) Jl =0.75m/s along the

diagonal line; (c) Jl =1.0m/s along the bisector line; (b) Jl =1.0m/s along the


Fig. 7.25 The relation between the turbulence anisotropy tensor b11 and b33 and the

local void fraction α in the duct center for different ducts. The black and

green symbols indicate the results in the small circular pipes and the large

xvi

ducts respectively.

Fig. 7.26 The relation between the turbulence anisotropy tensor b11 and b33 and the

local bubble size α in the duct center for different ducts. The black and

green symbols indicate the results in the small circular pipes and the large

ducts respectively.

Fig. 7.27 (a) Contribution of each term in the energy balance to the turbulence

anisotropy; (b) Schematic diagram of the generation of the turbulence

anisotropy of the single-phase turbulent flow in channel.

Fig. 7.28 Schematic diagram of grid-generated decaying homogeneous isotropic

turbulence.

Fig. 7.29 Schematic diagram of turbulence generation of the turbulent bubbly flow in

the core region where the turbulence is dominated by bubbles.

1

Chapter 1

Introduction

1.1 Motivation

Nowadays, in a wide variety of engineering systems the upward turbulent

bubbly flows play an increasingly important role such as boiling water and pressurized

water nuclear reactors (BWRs and PWRs), heat exchangers, petroleum transportation

systems and so on. Therein, accurate predictions of the flow characteristics are

essentially required for the design, process optimization and safety control. For example,

in the various subchannels of a nuclear fuel rod bundle of BWRs, the predictions of the

flow especially the void lateral distribution must be accurately carried out in order to

analyze the performance of the heat transfer such as local critical heat flux (CHF) which

is an important thermal-hydraulic phenomenon for the safety control. During the

transportation of the crude petroleum which is usually the two-phase or multiphase flow,

the predictions of the flow especially the drag friction are very crucial for the pipeline

design and operations such as the arrangement of the pump stations. Therefore, the

reliable two-phase flow models are in urgently needs based on the well understanding of

their physical processes.

So far, with the development of the experimental techniques and computational

fluid dynamics (CFD), numerous researches on the turbulent bubbly flow have been

carried out, on basis of which the understanding and modeling of turbulent bubbly flow

have been improved. However, the existence of the multi-deformable and moving

interfaces therein could induce the significant discontinuities of the fluid properties and

the complex flow field near the interface, which makes it much more complicated as

compared to the single-phase turbulent flow. Consequently, our current understanding

2

about the turbulent bubbly flows is still very limited due to its complicated properties,

and thus the predictive capability of the existing modeling is still far from the level of

the single-phase turbulent flow.

Due to its wide application in the various engineering systems and the lack of a

satisfactory knowledge about this complicated phenomenon, more researches related

with this phenomenon are necessary to be carried out in order to improve our physical

understanding and the existing models.

1.2 State-of-the-art of turbulent bubbly flow

In this section, the state-of-the-art of the upward turbulent bubbly flow will be

reviewed including the following topics:

(a) How were the previous researches on the turbulent bubbly flows carried out?

(b) What are the key issues in order to understand and model the turbulent

bubbly flow?

(c) What have we achieved based on the previous researches?

(d) What needs to be studied on the turbulent bubbly flow in the future?

1.2.1 Background

Generally, the researches to understand and model the turbulent bubbly flow

started from the single-phase turbulent flow which was considered as one of the most

complicated and unsolved physical problem in the last century and taken as the

background knowledge for the turbulent bubbly flow.

As for the single-phase turbulence, since the observation of the turbulence

phenomenon by Reynolds’s famous experiments in 1883, the innumerable researches

have been carried out from the experimental, theoretical and numerical point of views.

In general, the following topics were mainly focused on [1.1] including: (1) how the

turbulent flows behave; (2) what kinds of the fundamental physical processes involved;

(3) how to describe the turbulent flows quantitatively; (4) how to simulate and model

the turbulent flows to meet the requirements of the engineering application. Owing to

3

the contributors, our knowledge on single-phase turbulence has been improved

significantly in the past century.

As for the researches of the turbulent bubbly flows, the similar targets to that of

the single-phase turbulent flow are aimed at, which are solving the above four questions.

However, in the turbulent bubbly flows, there exist the multi-deformable and moving

interfaces which could also be coalesced to form new interfaces or broken into more

small interfaces [1.2]. Therefore, in addition to the general researches of the

single-phase turbulent flow, the bubbles behavior and bubbles’ effect are necessary to

be considered for the turbulent bubbly flow. Comparing to the single-phase turbulent

flow, the turbulent bubbly flow owns the different characteristics due to the existence of

the dispersed phase such as:

(a) Discontinuities of the fluid properties: During the mathematical description of

the turbulent bubbly flow, the general continuum fluid assumption used for

the single-phase flow becomes invalid because of the presence of the

multi-interfaces.

(b) Relative motion: Due to the density difference or the buoyancy effect, the

relative motions exist between the bubbles and the liquid phase.

(c) New characteristic scale: Additional characteristic scales related with the

bubble size DB are introduced on basis of the duct geometrical size DH and

eddy scales lv such as Kolmogorov scale and the integrated length scale le for

the single-phase turbulent flow.

(d) Phase interaction: Due to the relative motion and the new characteristic scale,

the flow characteristics in each phase could be influenced by the other phase

through the interfaces between them. For example, from the micro-scale point

of view, the presence of the interfaces could induce complex flow field near

the interface and consequently the complex flow field could modify the

interface characteristics such as bubble shape. For the turbulent bubbly flow

in the pipes, from the macro-scale point of view, the phase distribution could

be affected due the interaction between two phases. Consequently, the flow

4

field in the liquid phase such as velocities and turbulence distribution could

then be modified comparing to that of the single-phase flow. For a long time,

the understanding of the interfacial interaction, the phase distribution in the

turbulent bubbly flow and the turbulence modulation were very important

research topics and attracted numerous attentions.

(e) Complicated flow closure problem: With the interfacial interactions and

difficulties of the mathematical description, the flow closure problems of the

momentum and energy in the turbulent bubbly flow become more

complicated comparing to that of the single-phase turbulent flow. For

example, therein how to model the contributions of the phase interaction and

how to numerically solve the proposed models are both important research

topics.

Basically, for the turbulent bubbly flow, the existing researches from the

experimental, theoretical and numerical point of views were mainly focused on the

following questions aiming at accurate modeling of the turbulent bubbly flows.

(a) How does the turbulent bubbly flow behave? In addition to the general

researches of the single-phase turbulent flow, herein how does the bubbles

behave such as bubble distribution, deformation, coalescence and breakup

are also necessary to be considered.

(b) What kinds of the fundamental physical processes are involved in the

turbulent bubbly flow? In addition to the general researches of the

single-phase turbulent flow, herein what kind of the fundamental physical

processes responsible for the bubbles behaviors and the phase interactions

are also necessary to be considered.

(c) How to describe the turbulent bubbly flows quantitatively? In addition to the

general researches of the single-phase turbulent flow, herein how to solve the

problem induced by discontinuities of the fluid properties and how to

quantitatively describe the bubbles behaviors and phase interactions were

also necessary to be considered.

5

(d) How to simulate and model the turbulent bubbly flows to meet the

requirements of the engineering application? Additionally, how to model the

phase interaction based on the understanding the physical process are also

necessary to be considered.

In the following sections, what have we achieved about the above questions for

the turbulent bubbly flows will be reviewed based on the existing literatures. Firstly, the

establishment of the database of turbulent bubbly flow will be reviewed to answer the

first two questions mentioned above and validate the later proposed models. Then, the

development and implementation of the models for the turbulent bubbly flow will be

summarized to answer the latter two questions mentioned above. Finally, the unsolved

problems and the scope of the current dissertation will be given.

1.2.2 Review of the experimental and numerical databases

Currently, the databases for the turbulent bubbly flows are established based on

the experimental measurements and the direct numerical simulations (DNSs).

1.2.2.1 Experimental measurements

To understand the physical process and develop the model of the turbulent

bubbly flows, the detailed flow information such as the drag resistance, the temporal

and spatial evolutions of velocities and turbulence in two phases and the detailed

bubbles behaviors such as the bubble concentration, the bubble size, the bubble shape

and the bubbles motion are necessary at the same time. However, so far limited by the

current experimental techniques, for the turbulent bubbly flow, only several flow

parameters are available such as the statistical liquid phase velocities and turbulence

intensities, the void fraction, the bubble size and velocity, and the interfacial area

concentrations (IAC) and so on. The experimental databases on these parameters have

been established.

6

1.2.2.1.1 Database of turbulent bubbly flow

So far the databases of the turbulent bubbly flow have been established in the

following flow types:

(i) Uniform bubbly flow

The uniform flow is a simplest flow type for the understanding of the flow

characteristics and physical process and the developing the models. The studies on the

uniform turbulent flows such as the homogeneous turbulent flows generated by

homogeneous shear and the homogeneous isotropic turbulent flows generated by grids

have played an important role for the model development in the single-phase turbulent

flow [1.1]. As for the turbulent bubbly flow, so far Marie [1.3] and Lance et al.[1.4-1.5]

developed the measurements of the liquid-phase turbulence in a grid-generated and a

homogeneous shear-generated bubbly turbulent flow based on Laser-Doppler and

hot-film anemometry. Therein, the following liquid turbulent characteristics were

studied including the relation between the turbulent kinetic energy and the void fraction,

the isotropy level of the Reynolds stress tensor and the turbulent energy spectra.

According to their studies, two distinct regimes were found according to the void

fraction, which are a) for the low void fraction where hydrodynamic bubbles

interactions are negligible, the turbulent kinetic energy of the liquid could be considered

to be just the summation of the turbulent kinetic energy generated by the grid and the

motion of non-interactive bubbles; b) for the higher void fraction cases, due to the

bubbles mutual interactions, the bubbles transfer a greater amount of kinetic energy to

the liquid and the bubble-induced turbulence dominates the grid-generated turbulence.

In addition, during the above process, the isotropy of the turbulent field was found to be

not altered or improved by bubbles. The one-dimensional turbulent energy spectra of the

bubble-induced wake were found to be deviated from the classical power law exponents.

However, it is notable that in these experiments, the void fraction was very low less than

3% with the bubble diameter around 5mm. As for the other conditions, it still needs

further investigation.

7

(ii) Boundary layer flow

The boundary flow is another simple flow type for the understanding of the flow

characteristics and physical process and the developing the models for the single-phase

turbulent flow [1.1]. For the turbulent bubbly flow, so far there are very few researches

of this flow type, except Moursali et al.[1.6] developed the measurements in an upward

turbulent bubbly boundary layer along a vertical flat plate based on LDV. Therein, they

focused on the void fraction distribution, the wall shear stress, and the mean liquid

velocity profiles. According to their study, the lateral bubble migration toward to the

wall occurs depending on the bubble mean diameter and the void fraction similar to the

duct flow. In addition, the wall skin friction coefficient was observed to increase

because of the presence of the bubbles, which modifies the universal logarithmic law

near the wall. In this experiment, the measured location was mainly at 1m from the inlet

and the void fraction was less than 10% with the bubble diameter around 3.5mm.

However, as for the other conditions and measurements such as turbulence, it still needs

further investigation

(iii) Duct flow

According to the different duct geometries, several experimental databases of

the upward turbulent bubbly flows have been established in the small circular and large

circular pipes, and small non-circular and large noncircular ducts. The small and large

ducts were distinguished by the relation between the duct size DH and maximum bubble

size DMax ( gi

g 40 ) under the tested conditions, i.e., the small duct if DH <

DMax and otherwise large ducts [1.7]. The detailed literature survey of this flow type

could be found in [1.8]. In this section, the representative experimental studies and the

observations are briefly summarized.

In the small circular pipes with pipe diameter ranging from 15mm and 60mm,

the representative experimental studies were carried out by Serizawa [1.9-1.10],

Theofanous and Sullivan [1.11], Wang et al. [1.12], Liu [1.13-1.15], and Hibiki et al.

8

[1.16-1.17] by measuring the flow patterns and local flow distributions including

averaged liquid and gas velocities, void fraction, turbulence, bubble size, bubble

interfacial area concentration under several flow conditions. Therein, it was observed

that: for the upward turbulent bubbly flow in the small circular pipes, the pronounced

wall peak of the void fraction occurs for low area-averaged void fraction flow

conditions but core-peak occurs for the higher void fraction. The interesting

phenomenon was attributed to the lift force exerted on the bubbles which were

dependent on the bubble size [1.13]. With the wall peak of void fraction, the average

axial liquid velocity (Wl) profiles were observed to be more uniform in the core region

[1.9-1.17]. For higher void fraction cases, Theofanous and Sullivan [1.11] and Wang et

al. [1.12] observed slightly shift of the location of the maximum liquid velocity profile.

As for the turbulent structures, the radial distribution of the turbulence was observed to

be more uniform in the core region comparing to that of the single-phase flow.

Moreover, similar to the mirco-bubble case the turbulent drag reduction near the wall

was also observed under the low void fraction [1.18-1.19].

For the upward turbulent bubbly flow in the large diameter circular (pipe

diameter larger than 100 mm), Ohnuki and Akimoto [1.20] and Shawkat et al.

[1.21-1.22] established the comprehensive databases including the velocity field,

turbulent kinetic energy, void fraction, bubble size, bubble velocities and so on. The

experiments carried out [1.23-1.29] mainly measured the bubbles information including

void fraction, bubble size, bubble velocities and the interfacial area concentration

without the liquid phase flow field. According to the above databases, it was obtained

that the bubbles tend to migrate toward the pipe center earlier forming a core-peak of

the void fraction profile rather the wall-peak void distribution under the same flow

conditions even for the smaller bubbles with the bubble diameter 3-5mm [1.22], which

is different from those in small circular pipes. The wall-peak of the void fraction was

only observed for the cases with very low averaged void fraction less than 4% in the

experiments carried out by [1.20, 1.22, 1.24] in a 200mm pipe when the bubble diameter

was relatively small. No significant difference on the magnitude and shape for the

9

average liquid velocity was observed except the lower value in the near wall region

compared to that in the small circular pipe. Besides, as compared with that in the

smaller pipe, Prasser et al. [1.30] observed that under the same superficial velocities the

bubbles with larger diameter could move more freely and become more deformed in the

large pipes. The modifications of the distribution of the void fraction were attributed to

the deformation of the bubbles [1.31-1.33]. As for the liquid turbulence structure was

also considered to be one of the main mechanisms for the formation of the core-peak of

the void fractions in the large pipes. Lahey et al. [1.34] and Ohnuki and Akimoto [1.35]

suggested the strong turbulence diffusion effect on the bubbles is regarded as one reason

for the above phenomenon. However, more experimental databases of the liquid

turbulence are still necessary to validate these points.

For the upward turbulent bubbly flow in the small noncircular ducts, Lopez de

Bertodano et al. [1.36] established the comprehensive databases of the velocity fields,

Reynolds stress tensor and void fraction in the triangle duct. The other experiments in

the square and rectangular ducts by [1.37-1.41], mainly measured the void fraction.

According to the above databases, for noncircular ducts cases, it was observed that due

to the non-homogeneous geometrical boundary bubbles tend to be accumulated more in

the corner region, thus leading to the more significant corner peak of the void fraction

compared to the wall peak. The average liquid velocity was also found to be more

uniform comparing to the single-phase flow.

For the upward turbulent bubbly flow in the large noncircular ducts, recent

experiments done by Sun et al. [1.42] established detailed database for large vertical

square duct. Similar to the bubbly flow in other noncircular pipes [1.36-1.41], the

pronounced corner and wall peaks of void fraction were observed under the low

averaged void fraction cases. Due to the high local void fraction near the wall and the

corner the apparent peak of average liquid velocity was also found near corner and wall.

However, among the existing experiments in the circular small and large pipe and small

noncircular pipe no such strong velocity peak has been observed near the wall except

slightly peak near the wall [1.11-1.12]. In addition, the secondary flow in the cross

10

section observed in the single-phase turbulent flow in the noncircular ducts [1.43] may

have an effect on the flow characteristics. However, so far there are few experimental

measurements to show the existence and effect of the secondary flow.

1.2.2.1.2 Database of bubble behaviors

As mentioned above, the bubbles behavior is another key point to understand the

flow characteristics, the physical process and modeling of the turbulent bubbly flow in

the above flow types. So far, the basic databases of the bubble behaviors have been

established in the following flow types:

(i) Stagnant system

Therein, the important parameters including the terminal velocities VT, bubble

deformation E, and the drag coefficient CD were investigated based on the numerous

experiments of the single bubble [1.44]. The correlations of the above parameters were

proposed based on the following dimensionless numbers when it is dominated by

different effects such as the surface tension, inertial or viscosity effects.

2

Eol g BgD

,

4

2 3M

l l g

l

g

, Re l B R

B

l

D V

and

2

We l B RD V

. (1.1)

where Eötvös number Eo characterizes the relative importance of gravity to surface

tension; Morton number M the physical properties of systems; Reynolds number Re the

relative importance of inertial force to viscous force; and Weber number We the relative

importance of inertial force to surface tension. For the constant the fluid properties (the

constant M), the above dimensionless numbers solely depend on the bubble size DB,

since the relative velocity VR is also determined by the bubble size and the fluid

properties.

(ii) Uniform shear flow

Single-bubble experiments in the uniform shear flow were carried out by

Tomiyama et al.[1.33], where the bubble’s lateral migration was found to depend on the

11

bubble size DB or Eo. For example, the migration direction for the small bubble or Eo

was opposite to that for the large bubble or Eo. Besides, the well-known lift force

coefficient correlation was proposed, which will be given in the later section.

(iii) Upward turbulent bubbly flow

For the upward turbulent bubbly flow, Liu [1.13] studied the bubble size effect

on the flow characteristics in the small circular pipe by carefully controlling the inlet

bubble size DB. The lateral migration of bubbles and the flow regime transition were

found to be very sensitive to the bubble size. Under the low void fraction for the cases

of the bubble size DB < 5~6mm, the wall peak of the void fraction could be obtained,

while the core peak of the void fraction was observed for the large bubble size DB

>5~6mm. This was in consistent with the observation of Tomiyama’s single bubble

experiments [1.33]. In addition, the entrance length effect L/DH was also confirmed to

be an important parameter to determine the lateral void distribution due to the pressure

difference which could induce the bubble expansion. Along the axial direction, the

different flow regimes from the bubbly flow to slug flow are possible to be observed

due to the developing of the bubble coalescence and the bubble break-up.

1.2.2.1.3 Summary on experimental measurements

In summary, based on the above experimental databases, the following main

achievements were obtained for the turbulent bubbly flow:

(i) Bubble behaviors

For the upward turbulent bubbly flow in the ducts, there may exist the bubbles

lateral migration due to the additional force exerted on the bubbles from the liquid phase,

which could cause the phenomenon of the wall-peak, double peak or the core peak of

the void fraction. The lateral migration of the bubbles depends on the bubble size, the

bubble deformation, the duct size, and also the duct geometry. For the small bubbles,

under the low averaged void fraction the wall peak of the void fraction distribution

tends to be formed, while the core peak of the void fraction distribution tends to be

12

formed for the large bubbles. The duct size and geometry affects the void fraction under

which the transition from the wall peak to the core peak occurs earlier. Moreover,

increasing the averaged void fraction, the flow regime transition occurs which might

change from the dispersed bubbly flow to the annular flows, for which several flow

regime maps have been proposed [1.2].

(ii) Velocity fields in the liquid phase

The averaged liquid phase velocity and the turbulence fields were mainly

focused on. For the upward turbulent bubbly flow with the wall peak of the void

fraction, the liquid phase velocities were found to be more uniform in the core region.

For the turbulent bubbly flow with the core peak of the void fraction, the liquid phase

velocities were found to be steeper comparing to that of the single phase flow.

(iii) Turbulence modulation in the liquid phase

With the presence of the bubbles, the turbulence modulation occurs and

according to the above experiments, the observation could be summarized as follows:

(a) How the turbulence of the liquid phase was modulated strongly depends on

the bubble size and void fraction. Generally, with smaller particle or bubble

size smaller than the eddy size, the turbulence reduction occurs, while for

the large bubble size, the turbulence attenuation occurs [1.45]. However, for

the turbulent bubbly flow with relative large bubble size, under very low

void fraction cases such as less that 5%, the turbulence suppression could

also occur [1.19, 1.21]. For higher void fraction cases, the turbulence

enhancement occurs.

(b) The turbulent energy spectra in the turbulent bubbly flow were only

considered under very low void fraction. Therein, it was found that the

classical Kolmogorov power law (-5/3) was replaced by (-8/3) [1. 4, 1.21].

(c) Few detailed discussions of the turbulence anisotropy were carried out for

the upward turbulent bubbly flow, among which it was found to be

controversial. Theofanous and Sullivan showed the presence of the bubbles

13

didn’t alter the turbulent isotropy level [1.11]. Similar results were also

reported by Lance et al. [1.4, 1.5]. However, Wang et al. [1.12] showed the

bubbles modification of the local turbulence isotropy occurs which were

more anisotropic turbulence in the core region but more isotropic turbulence

near the wall [1.12].

1.2.2.2 Numerical simulations

Except experimental measurements, DNS has been proved to be a powerful tool

and played very important role in developing the turbulent models for the single-phase

turbulent flow. By directly solving the governing equations of the flow, DNS could

provide us all the scales of the turbulence without resorting any empirical closure

assumptions. Based on the VOF methods which are used to reconstruct the interface

structures, DNSs have been carried out for the turbulent flow with single bubble or

several bubble cases. However, for the cases with a large number of the bubbles, DNSs

started very late and are still very limited.

For the turbulent bubbly flow, currently Tryggvason group [1.46-1.56] has

constructed the numerical algorithms based on the finite difference method and the front

tracking of the bubble interface. With this numerical algorithm, a series of DNS studies

were performed for the homogeneous turbulent bubbly flow, vertical channel turbulent

bubbly flow and so on [1.46-1.56]. Therein, the detailed flow characteristics including

the flow fields, turbulence and bubbles behaviors were investigated.

For example, similar to the uniform turbulent bubbly flow carried out by Lance

and Bataille [1.4], Esmaeeli and Tryggvason [1.46-1.48] carried out DNSs of two- and

three- dimensional buoyant bubbles with regular array and freely array in the periodic

domains for the low and moderate Reynolds numbers. The turbulence generated therein

was compared the turbulence prediction based on the potential model [1.4]. The large

difference was found especially for the three-dimensional flow. In addition, the inverse

energy cascade of the turbulent structures was also noticed. Bunner and Tryggvason

[1.49-1.51] extended the previous DNS study [1.46-1.48] to more buoyant bubbles by

14

focusing on the bubbles behaviors including the rising velocity, the microstructures, the

dispersion and the fluctuation velocities of the bubbles and the flow fields such as the

bubble-induced pseudo-turbulence. Therein, the turbulent energy spectra showed a -3.6

power-law distribution at large wave numbers in consistent with Lance’s experimental

observations [1.4]. Moreover, the turbulence was found to be strongly anisotropic and

the anisotropy declined as the void fraction increased. The effects of the bubble

deformation on the flow properties were also investigated by Bunner and Tryggvason

[1.52], which showed a preference for pairs of the deformed bubbles to be vertically

aligned and for pairs of spherical bubbles to be aligned horizontal. Consequently, the

deformed bubbles tend to move toward the former bubbles’ wake, by which the stronger

velocity fluctuations could be induced by the deformed bubbles. Similar to the duct flow,

Lu et al. [1.53-1.55] carried out DNSs of the upward and downward turbulent bubbly

flows in a vertical channel with the low void fraction. Their numerical results [1.55]

showed that the bubble deformation rather than bubble size dominated the bubbles

lateral migration. In the upward flow, the deformed bubbles tend to migrate toward the

channel center, while the spherical bubbles tend to move the channel walls. Even with

relatively larger bubble size, the turbulent drag reduction was also obtained under low

void fraction when the bubble size was comparable to the buffer layer. Tryggvason

[1.56] addressed the multi-scale issues of the turbulent bubbly flow, and suggested the

multi-scale model for the turbulent bubbly flow based on the DNS databases.

Based on the DNS of the turbulent flow in a upward vertical channel, Bolotnov

[1.57] paid attention to the isotropy level of Reynolds stress distribution for the

single-phase and bubbly flow in order to address whether the isotropic turbulent

assumptions are suitable for the bubbly flows applications or not. His numerical results

showed that the presence of the bubbles improved the turbulence isotropy for low

Reynolds number flows, and didn’t alter for the moderate Reynolds number cases.

However, it is notable that the DNSs [1.57] only considered the extremely low void

fraction cases, say less than 3%.

15

1.2.2.3 Summary

So far, from the literatures, the numerical databases on the turbulent bubbly flow

have been established as summarized in Table 1.1, based on which our understanding on

the detailed fundamental processes and mechanisms of the turbulent bubbly flow were

improved. However, it is worth noting that in the above numerical simulations, only

tens at most hundreds of the bubbles were simulated due to the increasing of the

numerical difficulty especially the computer requirement, i.e., the numerical databases

were still limited in the low void fraction cases as compared to the real experiments.

There still exist some qualitatively controversial topics especially the bubble effects on

the turbulent anisotropy. In the simulations of the turbulent uniform bubbly flow by

Tryggvason groups [1.51-1.52], the turbulent behaves strongly anisotropic, while in

Bolotnov’s simulation in the vertical channel, the presence of the bubbles improve the

turbulence isotropy [1.57].

16

Table 1.1 Numerical databases based on DNSs of the turbulent bubbly flow.

Numerical Geometry Researchers Void fraction α

Bubble shape

S: Spherical;

D: Deformed

Note

2D Uniform flow Esmaeeli and Tryggvason

[1.46-1.48] 12.56% Eo = 1.0(S) -

3D Uniform flow

Esmaeeli and Tryggvason

[1.47-1.48]

Buner and Tryggvason

[1.49-1.52]

0%, 2%, 6%, 12%, 24%

Eo = 1.0(S)

Eo = 1.0(S), 5.0(D)

Regular bubble array

and Free bubbles

3D Vertical channel

Lu & Tryggvason

[1.53-1.55]

Bolotnov [1.57]

1.5%, 3.0%, 6.0%

0%, 1%

Eo = 0.45(S), 4.5(D)

Eo = 0.11(S)

-

-

17

1.2.3 Review of existing models for turbulent bubbly flow

Similar to the single-turbulent flow, though DNSs of the turbulent bubbly flow

could provide us the most detailed flow information, the requirements on the computer

memory and numerical time limited its application to the real industrial engineering

systems, especially for the large void fraction with a large number of the bubbles. The

models which could meet the requirements of the engineering application were more

efficient for the turbulent bubbly flows. This section will review the state-of-the-art of

the development and implementation of the models for the turbulent bubbly flows.

1.2.3.1 Model development

As mentioned before, in the turbulent bubbly flows the presence of the

multi-moving and deformable interfaces leads the flow to be significant discontinuous,

which brings the main numerical difficulties, i.e., resolving the fine interfacial structures.

During the model development of the turbulent bubbly flow the first important issue is

to eliminate the flow discontinuity and to obtain the macroscopic governing equations

which should satisfy the continuum fluid assumption based on the proper averaging

method.

Generally, according to the domain in the averaging method, the models for the

turbulent bubbly flow could be classified into two types, i.e., the one-dimensional

models and the multi-dimensional models. The one-dimensional models solve the

evolution of the cross-section averaged mean flow information such as the void fraction,

velocities, and pressure. The homogeneous equilibrium mixture model and the drift-flux

model by Zuber and Findlay [1.58] belong to this type. It requires less computer

memory and has been widely adopted in the past design calculations for the nuclear

reactor [1.59]. However, the models of this type require the prior knowledge of the local

distributions of phase, velocities and so on which are normally not known. As compared

to the one-dimensional model, the multi-dimensional models were derived based on the

time averaging, and can provide the local time-averaged flow information with the aid

of the powerful computer. Thus, to improve the models of the turbulent bubbly flow, the

18

numerical studies based on the multi-dimensional models are necessary.

According to the methods to consider the two phases during the averaging, the

models of the turbulent bubbly flow could be also classified into two types, i.e., the

mixture models and two-fluid models [1.2].

The mixture models consider the flow with both the continuous liquid phase and

the dispersed bubbles as a whole mixture, whose properties were determined by

coupling the properties of the two phases and each phase’s fraction. Consequently, the

conservation equations were established for the mass, momentum and energy based on

the whole mixture fluid. The homogeneous equilibrium mixture model is the simplest

example, which treats the fluid of the bubbly flow as a continuous mixture just like

single phase flow with modified the fluid properties. The relative motion between two

phases was neglected in the homogeneous equilibrium mixture model. Nowadays, the

drift-flux model proposed by Zuber and Findlay [1.58] is one widely used mixture

model, in which another equation for the relative motion between two phases was

included and the effects of two-phase interaction on the flow could be considered.

The two-fluid model proposed by Ishii [1.60] treats the two phases separately.

For each phase, the conservation equations including the mass, momentum and energy

were established based on the time averaging. Though the two-fluid model is more

complicated as compared with drift-flux model, however, in this way the more detailed

flow information such as the transfer processes of the mass, momentum and energy

between two phases could be predicted. Nowadays, the two-fluid model has been paid

lots of attentions, based on which the numerical predictions of the turbulent bubbly flow

were improved, and in the present dissertation this model will also be considered.

The basic governing equations of two-fluid model were shown in Eqs. (1.2) and

(1.3). Here, the flows were restricted to the adiabatic and incompressible two-phase

flows without the phase change.

0 kkkk

Dt

DV

(1.2)

19

kkkikkiwkik

kkkkkkkkkk

kkkk

pp

pDt

D

τMM

gvvVV

(1.3)

where the subscript-k refers to the liquid (l) and gas (g) flow; the subscript-i refers to the

effect of the interfaces; the overbar means the time-averaged quantities; the bold

variables are the vectors or tensors; k is void fraction of phase-k; Vk is the

corresponding phase velocity vector and could be separated by the mean value and the

fluctuations kkk vVV ; vk the fluctuating velocity vector; Dk /Dt indicates the

material derivative; ρk and μk are the density and viscosity of the corresponding phase-k;

pk is the static pressure of the corresponding phase-k; g is the gradational acceleration;

Mik is the interfacial force on phase-k; Mwk is the wall shear force on phase-k;

kkkkkk vvVτ ; kkk vv is Reynolds stress in the phase-k.

Comparing to the single-phase turbulent flow, the averaging process generates

the new interfacial transfer terms as shown by the last four terms in Eq. (1.3). The

interfacial term Mik related with the jump condition stratify Mil+ Mig = 0. In addition,

the turbulent Reynolds stresses also includes the phase interaction through the interfaces.

To improve the model of the turbulent bubbly flow the current work was mainly focused

on the establishment and validation of the constitutive relations for the interfacial

transfer terms and the turbulent Reynolds stresses based on the established experimental

databases. Some attentions were also paid to the numerical algorithms to solve the

above equations. In the following, the state-of-the-art of the constitutive relations for the

two-fluid model will be reviewed in brief.

1.2.3.1.1 Interfacial transfer terms

During the model development, the interfacial term Mik was decomposed into

the drag force d

ikM and the non-drag forcend

ikM ,

nd

ik

d

ikik MMM (1.4)

20

The drag force d

ikM is generally expressed by the following standard form in

terms of the drag coefficient CD, the relative velocity Vr between two-phases and the

interfacial area concentration ai of the bubbles [1.60]:

irrlD

d

ig aC VVM 21 . (1.5)

The attentions were paid to establish the correlations for CD and ai. Several

correlations for CD have been established based on the experiments of the single solid

particle, droplets, or bubbles in the infinite systems [1.44]. Ishii and Chawla [1.61]

developed the correlation of CD for the multi-particle systems considering the flow

regimes and the characteristics of the particles.

As for the interfacial area concentration ai, the simplest expression is given in

terms of the Sauter mean diameter DSM and the void fraction α assuming the bubbles to

be spherical shape: ai=6/DSM, where DSM is the volume equivalent bubble diameter of

spheres. Ishii and Mishima [1.62] developed the correlations of the interfacial area

concentration ai considering the flow regimes as shown in Eq. (1.6).

6

64.5 11 1

For bubbly flow

For slug or churn turbulent flow

SM

gs gs

H gs H gs

D

i

D D

a

. (1.6)

As for the non-drag forcend

ikM , the following forces were usually considered

according to the single-particle theory.

Lahey et al. [1.63] studied the constitutive equations of the virtual mass force for

the bubbly flow. They found that the introduction of the virtual mass force V

igM term

could improve the numerical stability and efficiency, and thus was important to the

two-fluid model. Ishii and Mishima [1.62] summarized and gave the correlations for

V

igM in term of the virtual mass coefficient CVM and the acceleration of the relative

velocities according to the flow regimes as

V rig VM l r r

DC

Dt

VM V V , (1.7)

21

and

12

1 2 For bubbly flow

1

1 /5 0.66 0.34 For slug flow

1 / 3

VM

b b

b b

CD L

D L

(1.8)

According to the analysis of the single-spherical particle moving in the viscous

shear flow [1.44], the particle could receive the lift force L

igM from the liquid phase

perpendicular to the flow direction. Similarly, for the turbulent bubbly flow with the

relative velocity Vr and the liquid phase velocity gradient lV , the lift force exerted on

the bubbles was also considered and expressed in term of the lift force coefficient CLM,

the relative velocity Vr, and the liquid phase velocity gradient lV as

L

ig L l r lC M V V . (1.9)

The lift force played a very important role to the prediction of the bubble lateral

distributions [1.10-1.17]. Attentions have been paid to determine the lift force

coefficient CLM based on the single bubble experiments and simulations [1.33, 1.34,

1.44], which was found to be related with the bubble size and deformation. According to

Tomiyama’s experiments [1.33], the lift force coefficient CLM was given as follows:

H

H

min 0.288tanh 0.121Re , Eo For Eo 4

Eo For 4 Eo 10

0.27 For 10 E

H

LM H

f

C f

Ho

(1.10)

where 3 2

H H H HEo 0.00105Eo -0.0159Eo -0.0204Eo 0.474f ; 2

HEoj g HBg D

;

HBD the maximum horizontal bubble dimension.

The constitutive relations of the turbulent dispersion force T

igM exerted on the

bubbles were derived by Lahey [1.34], Bertodano [1.65-1.66], and Burns [1.67]

respectively using different methods as follows:

22

13

4 0.9 1

From Lahey [1993]

From Bertodano [2006]

From Burns [2004]

t

b

t

T l

T

ig T l

T D r ld

C k

C

C C

M

V

(1.11)

where CT is the turbulent diffusion coefficient; k is the turbulent kinetic energy; υt is the

eddy viscosity.

Antal et al. [1.68] introduced and derived the wall-lubrication force w

igM

exerted on the bubble into the two-fluid model to prevent the bubble from touching the

wall based on the analysis of single spherical bubble moving in the laminar flow. The

lateral distribution near the wall could be better predicted with this force w

igM as

expressed in the following form:

22w

ig l W r wdCM V n (1.12)

where wn is the wall-normal unit vector; CW is the wall force coefficient which is

related with the distance y to wall and expressed as

3 3

2 2

2

4 4

2

0.104 0.06 0.147 From Antal [1991]

Eo From Tomiyama [1998]

0.005 0.05 From ANSYS [2010]

B

B B

H

B

D

r w y

D D

W y D y

D

y

C y F

V n

(1.13)

with H

H

exp -0.933Eo 0.179 For 1 Eo 5Eo

0.007Eo 0.04 For 5 Eo 33F

.

The last two terms in Eq. (1.3) indicate the interfacial pressure and shear stress

terms. Lahey [1.70] and Bertodano et al. [1.71] considered the same pressure in both

phases pk=p, τik=τk and introduced the following constitutive relations for interfacial

pressure and shear stress terms:

23

2P

ig ig g p l rp p C M V (1.14)

S

ig ig g M τ τ (1.15)

where Cp is the pressure coefficient and 0.25 for non-interacting spherical bubbles based

on the inviscid result. Bertodano et al. [1.66] discussed the non-spherical cases where

Cp > 0.25 due to the wake effect behind the bubbles.

Though several experimental and analytical efforts were carried out to improve

the above correlation coefficients [1.2], so far the physical mechanism of the above

forces are still poorly understood. In addition, it is also notable that the aforementioned

correlations were mainly proposed based on the single particle case, for the bubbly flow

with many bubbles and bubble-bubble interaction further efforts are still necessary.

1.2.3.1.2 Turbulent Reynolds stresses

Turbulence term T

k k k kτ v v in the above momentum equations is one of the

most important key terms for the turbulent bubbly flow, which determines the prediction

of the distributions of the velocities, void fraction, and interfacial term. Turbulent

models for the single-phase turbulence have been widely studied and could be found

everywhere in the textbooks about the turbulence [1.1]. However, for turbulent bubbly

flow, the understanding and modeling of this term are still in an initial stage due to the

complicated physical process. Therein, the following mechanisms for the turbulence

generation coexist such as the general shear-induced turbulence, the interaction between

the shear-induced turbulent structures and the bubbles, the bubbles-induced turbulence

and its interaction between the different turbulent structures [1.5, 1.9]. So far, the

development of the turbulent models for the turbulent bubbly flow was mainly carried

out based on the analogy of the single-phase turbulent models.

Based on the phenomenological modeling where the eddy viscous assumption

for the turbulent shear stress / 2T T

S k l t l l τ V V was also adopted, Sato

and Sadatomi [1.72], Drew and Lahey [1.73] and Kataoka and Serizawa [1.74] adopted

24

the zero or one equation model to the turbulent bubbly flow to determine the turbulent

shear stress by adding the bubble-induced turbulence. Therein, the attentions were also

paid to the determination of the eddy viscosity t and the mixing length lm.

2

From Sato, 1981

From Drew and Lahey, 1981

2 From Kataoka and Serizawa, 1995

3

l s b

T

S l m

l s B

dV

dr

dV dVl

dr dr

k dVl l

dr

(1.16)

where s and b are the eddy viscosity induced by shear and bubbles, in Sato and

Sadatomi [1.72] b b B rC D V ; sl and bl are the mixing lengths of the shear

induced turbulence and bubble induced turbulence respectively; s and sl were

decided based on the Prandtl mixing theory. However, the determinations of the eddy

viscosity and mixing length are strongly dependent on their experimental data, and so

far the applications of the above models were still limited and need further validation.

Theofanous and Sullivan [1.11] and Wijngaarden [1.75] theoretically and

statistically discussed the bubble-induced turbulence or the pseudo-turbulence by

comparing the turbulent intensities before and after the bubbles injection. Based on the

potential theory Wijngaarden [1.75] derived the pseudo turbulent Reynolds stress for the

rising bubbles in the laminar flow as

23 1

20 20B r r rI

τ V V V , (1.17)

and for the uniform turbulent flow,

2

22 0

26 0rE

r

uk M f f a

VV

(1.18)

For the bubbly flow in the pipe, Theofanous and Sullivan [1.11] estimated the

total turbulence based on the mean flow velocity and the wall shear stress as

25

2

11 1 1 1 1 1.5

2 4

g H Hf

c l c B

vv D g Dc

V V D

. (1.19)

Several more general and accurate turbulent models for turbulent bubbly flow

have also been developed on basis of the two-equation model or Reynolds stress model

of the single phase turbulent flow [1.1]. Kataoka et al. [1.76] derived the basic

equations for the turbulent Reynolds stress and dissipation for the two-phase flow,

where the expressions of the interfacial terms were given and discussed. So far,

according to the author’s knowledge, the following three different methods were used to

consider the turbulence.

Lahey [1.70] developed η-ε model for the turbulent stresses in the liquid phase of

the turbulent bubbly flow by mimicking the single-phase η-ε model as follows:

2l l l S l i

D kC

Dt

vv vv vv P Φ I S (1.20)

1 2i

l l l l B

e

D SC C P C C

Dt k k k t

vv (1.21)

P, Φ, ε and I are the turbulence production, redistribution, dissipation and unit

tensor which are the same as the single-phase turbulent flow. Cε, Cε1, Cε2 and CεB are the

model constants. For the turbulent bubbly flow, the additional interfacial tensor Si and

/B i eC S t are the bubble induced turbulence generation and dissipation;

tracei iS S ; te is the characteristic time of the eddies, and the choice of te was

discussed by Rzehak and Krepper [1.77].

By 12

tracek vv , the above Reynolds stress model become the k-ε model for

the turbulent bubbly flow in the following form:

l l l t l i

Dk k P S

Dt . (1.22)

Several studies were paid to the constitutive relations of the additional interfacial

26

tensor Si and the additional interfacial dissipation term /B i eC S t [1.77].

Bertodano et al. [1.71] proposed another type of η-ε model considering the

transport equations for both the shear-induced turbulent and the bubble-induced

turbulence:

l S l l t S l S S

Dk k P

Dt . (1.23)

BBam

b

lBtllBl kk

tkk

Dt

D

. (1.24)

1 2

l t S Sl S l S l S

t

DC P C

Dt k k

. (1.25)

In addition, the turbulent shear stress was expressed as

2

/ 23

T

l t S l S B Bk k τ V V A A (1.26)

where the eddy viscosity

2

t b B r

kC C D

V , and AS and AB are the tensors to

show the turbulence anisotropy for the sear-induced turbulence and bubble-induced

turbulence respectively.

Similar to Lahey [1.70] and Bertodano et al. [1.71], Chahed et al. [1.78-1.79]

proposed another turbulent model by decomposing the turbulent Reynolds stress into

turbulent dissipative part due to the velocity gradient and the pseudo-turbulent

non-dissipative part due to the interfacial drag. Therein, the turbulent anisotropy was

considered by introducing the bubble characteristics time following the experimental

results and initial model of Lance [1.4-1.6].

On the other hand, as observed in the experimental and numerical databases, the

bubbles size and deformation could strongly affect the flow characteristics. The

aforementioned models were developed based on a prior knowledge of the bubbles size

and deformation such as the averaged bubble Sauter mean diameter DSM. However, in

the real systems, there exist a broad range of the bubble sizes, and the bubbles could

27

experience the coalescence with other bubbles and also breakup, which were not shown

in the above models at all. Recently, to improve the models for the turbulent bubbly

flow, the attentions were also paid to the bubble behaviors such as distributions of the

bubble size or interfacial area concentration (IAC) [1.2]. Generally, the models to

predict the bubble behaviors are based on Boltzman statistical averaging method and

Boltzman equation for the dispersed bubbles. Correspondingly, the dispersed gas phase

was separated into several groups according to the interested properties of the dispersed

bubbles such as bubble size and IAC. For each group of the dispersed phase, the

conservations equations of the mass, momentum, and energy were established

considering the birth and death of the each group. For the continuous phase, the

interfacial effects from the dispersed phase are the sum of all groups. With this method,

it aimed at predicting the behaviors of the dispersed gas phase such as the distributions

of the bubble size or interfacial area concentration which are crucial to the flow

characteristics. Currently, the population balance model [1.80], the one-group and

two-group interfacial transport model [1.2] and multi-fluid model [1.81] belong to the

model of this type. Therein, the attentions were mainly paid to model the birth and death

of the each group which depend on the bubble coalescence and breakup. The

corresponding detailed reviews could be found in [1.82-1.83]. Due to the complicated

nature of the bubble coalescence and breakup, these models are still in an initial stage

and the more efforts are required in the future work.

1.2.3.2 Model implementation and validation

As mentioned in Section 1.2.2, the current experimental databases of the

turbulent bubbly flow were mainly established in the following flow types: uniform

bubbly flow, bubbly boundary layer flow, the bubbly flow in the small and large circular

pipes and small and large noncircular ducts. With the aid of these databases, the

implementations and validations of the two-fluid model were carried out such as in

[1.70-1.85]. At this stage, most of the attentions were focused on the prediction of the

phase distributions by considering the interfacial terms in the momentum transport

28

equations [1.70-1.74]. However, for the prediction of the characteristics of the turbulent

structures such as intensities and anisotropic properties, even several models have been

proposed [1.75, 1.78-1.79], they were limited for very low void fraction or not

satisfactory due to our limited knowledge on the bubble-induced turbulence. Moreover,

the previous implementation and validation of models were mainly carried out in the

small circular pipes [1.70-1.71, 1.84]. For the bubbly flow in the large circular and

noncircular ducts where the flow were strongly affected by the duct geometry and

showed different characteristics, there were still very few detailed validation work

[1.34-1.35, 1.85]. In addition, with the above models only the drag enhancement could

be predicted due to the superposition of the shear-induced turbulence and

bubble-induced turbulence, whereas the turbulent drag reduction were also observed in

the experiments [1.19]. Therefore, whether the models proposed and validate before are

suitable for the different conditions still needs further studies based on the new

experimental databases. And the new models for other flow properties such as

turbulence are also required.

1.2.4 Summary on review of databases and models

In section 1.2.2-1.2.3, the state-of-the-art of the turbulent bubbly flow was

briefly reviewed including the existing databases and models. So far, the experimental

databases of the turbulent bubbly flow have been developed for several flow types

including uniform bubbly flow, circular pipe flow, and noncircular duct flow. The flow

characteristics such as phase distribution and turbulence were mainly analyzed and

modeled for the turbulent bubbly flow in the small circular pipes and noncircular ducts.

However, for the turbulent bubbly flow in the large and non-circular ducts which could

be often met in the engineering applications, it is still lack of the experimental database

and the model evaluations. The previous developed models were focused on the

prediction of the phase distributions, but about the turbulent characteristics such as

intensities, turbulence generation process and anisotropic properties the current

understandings are still far from the satisfactory and limited for very low void fraction.

29

1.3 Scope and layout of the dissertation

On basis of the background of the turbulent bubbly flow mentioned above, the

present dissertation is aimed at the following aspects to improve the understanding and

modeling of the turbulent bubbly flow especially in the large noncircular duct:

(1) Evaluating the models proposed for the turbulent bubbly flow in the small

circular and noncircular ducts to that in the large noncircular duct;

(2) Further analyzing the flow characteristics of the turbulent bubbly flow in

the large noncircular ducts especially the liquid phase velocities and

turbulence;

(3) Improving the current models for the prediction of the turbulent bubbly

flow in the large noncircular ducts especially for the turbulence structures

based on the proposed analysis above.

The dissertation is composed by 8 chapters and arranged as follows: Chapter 1

mainly introduces the motivation, the current research status and the purpose of this

study. Chapter 2 describes the experimental database including the experimental

apparatus, flow conditions and some experimental results. Chapter 3 describes the

numerical methodologies based on the two-fluid model for the turbulent bubbly flow in

the large square duct, numerical results and discussions. To improve the modeling of the

turbulent bubbly flow in the large noncircular ducts, the further analysis and modeling

improvement are carried out for flow characteristics in the later four chapters. Chapter 4

discusses the developing process in the turbulent bubbly flow. Chapter 5 is focused on

the characteristics and modeling of the liquid phase velocities. Chapter 6 proposes a

new correlation for the bubble deformation and describes the numerical validation of the

void fraction. Chapter 7 describes the turbulence modulation due to the presence of the

bubbles. Finally, the summary of this dissertation and the recommendation of the future

work will be given in Chapter 8.

30

Reference

[1.1] Pope, S. B., 2000, Turbulent flows, Cambridge University Press.

[1.2] Ishii, M., Hibiki, T., 2011, Thermo-fluid dynamics of two-phase flow. 2nd ed.

New York: Springer.

[1.3] Marié, J. L., 1983, Investigation of two-phase bubbly flows using Laser Doppler

Anemometry, PCH Physico Chemical Hydrodynamics 4, 103-118.

[1.4] Lance, M. , Bataille, J., 1991, Turbulence in the liquid phase of a uniform

bubbly air-water flow, J. Fluid Mechanics 222, 95-118.

[1.5] Lance, M. , Bataille, J., 1991, Homogeneous turbulence in bubbly flows, J.

Fluids, Engineerig 113, 295-300.

[1.6] Moursali, E., Bataille, J., Marié, J. L., 1995, An upward turbulent bubbly

boundary layer along a vertical flat plate, Int. J. Multiphase Flow 21, 107-117.

[1.7] Isao, K., Mamoru, I., 1987, Drift flux model for large diameter pipe and new

correlation for pool void fraction, Int. J. Heat Mass Transfer 30, 1927–1939.

[1.8] Sun, H. M., 2014, Study on upward air-water two-phase flow characteristics in a

vertical large square duct, Doctor Thesis, Kyoto University.

[1.9] Serizawa, A., 1974, Fluid dynamics characteristics of two-phase flow, Doctor

Thesis, Kyoto University.

[1.10] Serizawa, A., Kataoka, I., Michiyoshi, I., 1975, Turbulence structure of air -

water bubbly flow—II. local properties, Int. J. Multiphase Flow 2, 235–246.

[1.11] Theofanous, T. G., Sullivan, J., 1982, Turbulence in two-phase dispersed flows,

J. Fluid Mechanics 116, 343-362.

[1.12] Wang, S., Lee, S., Jones, O., Lahey, R., 1987, 3-D turbulence structure and

phase distribution measurements in bubbly two-phase flows, Int. J. Multiphase

Flow 13, 327–343.

[1.13] Liu, T. J., 1993, Bubble size and entrance length effects on void development in

a vertical channel, Int. J. Multiphase Flow 19, 99–113.

[1.14] Liu, T. J., Bankoff, S. G., 1993, Structure of air-water bubbly flow in a vertical

31

pipe—I. liquid mean velocity and turbulence measurements, Int. J. Heat Mass

Transfer 36, 1049–1060.


pipe—I. bubble void fraction, bubble velocity and bubble size distribution, Int. J.

Heat Mass Transfer 36, 1061–1072.

[1.16] Hibiki, T., Ishii, M., 1999, Experimental study on interfacial area transport in

bubbly two-phase flows, In. J. Heat Mass Transfer 42, 3019–3035.

[1.17] Hibiki, T., Ishii, M., Xiao Z., 2001, Axial interfacial area transport of vertical

bubbly flows, In. J. Heat Mass Transfer 44, 1869–1888.

[1.18] Michiyoshi, I., Serizawa, A., 1986, Turbulence in two-phase bubble flow, Nucl.

Eng. Des. 95, 253-267

[1.19] Serizawa, A., Kataoka, I., 1990, Turbulence suppression in bubbly two-phase

flow, Nucl. Eng. Des. 122, 1–16.

[1.20] Ohnuki, A., Akimoto, H., 2000, Experimental study on transition of flow pattern

and phase distribution in upward air–water two-phase flow along a large vertical

pipe, Int. J. Multiphase Flow. 26, 367–386.

[1.21] Shawakat, M. E., Ching, C. Y., Shoukri, M., 2007, On the liquid turbulence

energy spectra in two-phase bubbly flow in a large diameter vertical pipe, Int. J.

Multiphase Flow 33, 300-316.

[1.22] Shawakat, M. E., Ching, C. Y., Shoukri, M., 2008, Bubble and liquid turbulence

characteristics of bubbly flow in a large diameter vertical pipe, Int. J.


[1.23] Sun, X., Smith, T. R., Kim, S., Ishii, M., Uhle, J., 2003, Interfacial structure of

air-water flow in a relatively large pipe. Exp. Fluids 34, 206-219.

[1.24] Shoukri, M., Hassan, I., Gerges, I., 2003, Two-phase bubbly flow structure in

large-diameter vertical pipes, The Canadian J. Chem. Eng. 1 205-211.

[1.25] Shen, X., Mishima, K., Nakamura, H., 2005, Two-phase phase distribution in a

vertical large diameter pipe, Int. J. Heat Mass Transfer 48, 211–225.

32

[1.26] Schlegel, J.P., Sawant, P., Paranjape, S., Ozar, B., Hibiki, T., Ishii, M., 2009,

Void fraction and flow regime in adiabatic upward two-phase flow in large

diameter vertical pipes, Nucl. Eng. Des. 239, 2864–2874.

[1.27] Smith, T.R., Schlegel, J.P., Hibiki, T., Ishii, M., 2012, Two-phase flow structure

in large diameter pipes, Int. J. Heat Fluid Flow 33, 156–167.

[1.28] Shen, X., Hibiki, T., Nakamura, H., Developing structure of two-phase flow in a

large diameter pipe at low liquid flow rate, Int. J. Heat Fluid Flow. 34 (2012)

70–84.

[1.29] Schlegel, J.P., Miwa, S., Chen, S., Hibiki, T., Ishii, M., 2012, Experimental

study of two-phase flow structure in large diameter pipes, Exp. Thermal Fluid

Sci. 4, 12–22.

[1.30] Prasser, H. M., Beyer, M., Boettger, A., Carl, H., Lucas, D., Schaffrath, A.,

Schuetz, P., Weiss, F. P., Zschau, J., 2005, Influence of the Pipe Diameter on the

Structure of the Gas-Liquid Interface in a Vertical Two-Phase Pipe Flow, Nucl.

Tech. 152, 3-22

[1.31] Esmaeeli, E. A., Tryggvason, G., 1994, Numerical simulations of rising bubbles

in bubbles dynamics and interface phenomena, Kluwer Academic, Dordrecht.

[1.32] Ervin, E. A., Tryggvason, G., 1997, The rise of bubbles in a vertical shear flow.

J. Fluids Eng. 119, 443-449.

[1.33] Tomiyama, A., Tamai, H., Zun, I., Hosokawa, S., 2002, Transverse migration of

single bubbles in simple shear flows, Chem. Eng. Sci. 57, 1849-1858.

[1.34] Lahey, Jr R. T., Bertodano, de M. L., Jones, Jr O. C., 1993, Phase distribution in

complex geometry conduits, Nucl. Eng, Des. 141, 177-201.

[1.35] Ohnuki, A., Akimoto, H., 2001, Model development for bubble turbulent

diffusion and bubble diameter in large vertical pipes. J. Nucl. Sci. Technol. 38,

1074-1080.

[1.36] Bertodano, de M. L., 1992, Turbulent bubbly two-phase flow in a triangular duct.

Doctoral Thesis, Resselaer Polytechnic Institute.

33

[1.37] Launder, B.E., Ying, W.M., 2006, Secondary flows in ducts of square

cross-section, J. Fluid Mech. 54, 289.

[1.38] Ohba, K., Yuhara, T., 1982, Study on vertical bubbly flows using laser doppler

measurements: 2nd report, turbulence structure in square duct flow. Trans. Japan

Society of Mech. Eng. 48, 78-87

[1.39] Ohba, K., Yuhara, T., 1985, Matsuyama H. Study on vertical bubbly flows using

laser doppler measurements: 3rd report, simultaneous measurement of bubble

and liquid. Trans. Japan Society Mech. Eng. 51, 4194-4200.

[1.40] Zun, I., 1990, The mechanism of bubble non-homogeneous distribution in

two-phase shear flow. Nucl. Eng. Des. 118, 155-162.

[1.41] Sadatomi, M., Sato, Y., Saruwatari, S., 1982, Two-phase flow in vertical

noncircular channels. Int. J. Multiphase Flow 8, 641–655.

[1.42] Sun, H. M., Kunugi, T., Shen, X. Z., Wu, D. Z. and Nakamura, H., Upward

air-water bubbly flow characteristics in a vertical square duct, J. Nucl. Sci. Tech.,

Vol. 51, 267–281(2014).

[1.43] Hoagland, L. L., 1960, Fully Developed Turbulent Flow in Straight Rectangular

Ducts - Secondary Flow, Its Cause and Effect on the Primary Flow, Ph.D. thesis,

Department of Mechanical Engineering, MIT.

[1.44] Clift, R., Grace, J.R., Weber M.E., 1972, Bubbles, drops, and particles, Science.

[1.45] Gore, R. A., Crowe, C. T., 1989, Effect of particle size on modulating turbulent

intensity, Int. J. Multiphase Flow 15, 279-285.

[1.46] Esmaeeli, A., Tryggvason, G., 1996, An inverse energy cascade in

two-dimensional low Reynolds number bubbly flows, J. Fluid Mech. 314,

315-330

[1.47] Esmaeeli, A., Tryggvason, G., 1998, Direct numerical simulations of bubbly

flows Part 1. Low Reynolds number arrays, J. Fluid Mech. 377, 313-345.

[1.48] Esmaeeli, A., Tryggvason, G., 1999, Direct numerical simulations of bubbly

flows Part 1. Moderate Reynolds number arrays, J. Fluid Mech. 385, 325-358.

34

[1.49] Bunner, B., Tryggvason, G., 1999, Direct numerical simulations of

three-dimensional bubbly flows, Phys. Fluids 11, 1967-1969.

[1.50] Bunner, B., Tryggvason, G., 2002, Dynamics of homogeneous bubbly flows

Part 2: Rise velocity and microstructure of the bubbles, J. Fluid Mech. 466,

17-52.


Part 2: velocity fluctuations, J. Fluid Mech. 466, 53-84.

[1.52] Bunner, B., Tryggvason, G., 2003, Effect of bubble deformation on the

properties of bubbly flows, J. Fluid Mech. 495, 77-118.

[1.53] Lu, J., Fernández, A., Tryggvason, G., 2005, The effects of bubbles on the wall

drag in a turbulent channel flow, Phys. Fluids 17, 095102.

[1.54] Lu, J., Tryggvason, G., 2006, Numerical study of turbulent bubbly downflows in

a vertical channel, Phys., Fluids 18, 103302.

[1.55] Lu, J., Tryggvason, G., 2008, Effect of bubble deformability in turbulent bubbly

upflow in a vertical channels, Phys. Fluids 20, 040701.

[1.56] Tryggvason, G., 2013, Multiscale issues in DNS of multiphase flows, 8th

International Conference on Multiphase Flow, 2013, Jeju, Korea.

[1.57] Bolotnov, Igor A., 2013, Influence of bubbles on the turbulence anisotropy, J.

Fluids Eng. 135, 051301.

[1.58] Zuber, N., Findlay, J. A., 1965, Average volumetric concentration in two-phase

flow systems, J. Heat Transfer, 453-468.

[1.59] Wheeler, C. L., 1986, COBRA-NC: a thermal-hydraulic code of transient

analysis of nuclear reactor components: equations and constitutive models.

NUREG/CCR-3262.

[1.60] Ishii, M., 1975, Thermo-fluid Dynamic Theory of Two-phase Flow, Eyrolles.

[1.61] Ishii, M., Chawla, T. C., Local drag laws in dispersed two-phase flows, Argonne

Nation Lab Report, ANL-79-105.

[1.62] Ishii, M., Mishima, K., 1984, Two-Fluid model and hydrodynamics constitutive

relations, Nucl. Eng. Des. 82, 107-126.

35

[1.63] Lahey, Jr. R.T., Cheng, L. Y., Drew, D. A.,, Flaherty, J. E., 1980, The effects of

virtual mass on the numerical stability of accelerating two-phase flows, Int. J.

Multiphase flow 6, 281-294.

[1.64] Tomiyama, A., Sou, A., Zun, I., Kanami, N., Sakaguchi, T., 1995, Effects of

Eötvös number and dimensionless liquid volumetric flux on local motion of a

bubble in a laminar duct flow, Advance In Multiphase Flow, Elsevier, 3-15.

[1.65] Bertodano, Lopez M., 1998, Two fluid model for two-phase turbulent jets, Nucl.

Eng. Des. 179, 65-74.

[1.66] Bertodano, Lopez M., Sun, X., Ishii, M., Ulke, A., 2006, Phase distribution in

the cap bubble regime in a duct, J. Fluids Eng. 977-1001.

[1.67] Burns, D., Frank, T., Hamill, I., Shi, J. M., 2004, The Favre averaged drag

model for turbulent dispersion in Eulerian multi-phase flow. In: 5th International

Conference on Multiphase Flow, Yokohama, Japan, 392.

[1.68] Antal, S.P., Lahey, Jr R. T., Flaherty, J. E., 1991, Analysis of phase distribution

in fully developed laminar bubbly two-phase flow, Int. Multiphase Flow 13,

309-326.

[1.69] Tomiyama, A., 1998, Struggle with computational bubble dynamics. In: 3rd

International Conference on Multiphase Flow, ICMF’98, Lyon, France.

[1.70] Lahey Jr. R.T., 1990, The analysis of phase separation and phase distribution

phenomena using two-fluid models, Nucl. Eng. Des. 122, 17-40.

[1.71] Bertodano, Lopez M., Lahey, Jr. R.T., Jones, O. C., 1994, Phase distribution in

bubbly two-phase flow in vertical ducts, Int. J. Multiphase Flow 20, 805-818.

[1.72] Sato, Y., Sadatomi, M., 1981, Momentum and heat transfer in two-phase bubble

flow—I. Theory, Int. J. Multiphase Flow 7, 167-177.

[1.73] Drew, D. A., Lahey, Jr. R.T., 1982, Phase distribution mechanisms in turbulent

low-quality two-phase flow in a circular pipe, J. Fluid Mech. 117, 91-106.

[1.74] Kataoka, I., Serizawa, A., 1995, Modeling and prediction of turbulence in

bubbly two-phase flow, Proc. 2nd

Int. Conf. Multiphase Flow, Kyoto,

MO2:11-16

http://www.sciencedirect.com/science/article/pii/0301932281900033

36

[1.75] Wijngaarden, van L., 1998, On pseudo turbulence, Theoret. Comput. Fluid

Dynamics 10, 449-458.

[1.76] Kataoka, I., Serizawa, A., 1989, Basic equations of turbulence in gas-liquid

two-phase flow, Int. Multiphase Flow 5, 843-855.

[1.77] Rzehak, R., Krepper, E., 2013, Bubble-induced turbulence: Comparison of CFD

models, Nucl. Eng. Des. 258, 57-65.

[1.78] Chahed, J., Roig, V., Masbernat, L., 2003, Eulerian-Eulerian two-fluid model for

turbulent gas-liquid bubbly flows, Int. J. Multiphase Flow 29, 23-49.

[1.79] Bellakhel, G., Chahed, J., Masbernat, L., 2004, Analysis of the turbulence

statistics and anisotropy in homogeneous shear bubbly flow using a turbulent

viscosity model, Journal of Turbulence 5, 036

[1.80] Das, A. K., Das, P. K., 2010, Modeling bubbly flow and its transitions in vertical

annuli using population balance technique, Int. J. Heat Fluid Flow 31, 101-104.

[1.81] Politano, M. S., Carrica, P. M., Converti, J., 2003, A model for turbulent

polydisperse two-phase flow in vertical channels, Int. J. Multiphase Flow 19,

1153-1182.

[1.82] Liao, Y., Lucas, D., 2009, A literature review of theoretical models for drop and

bubble breakup in turbulent dispersions, Chem. Eng. Sci. 64, 3389-3406.

[1.83] Liao, Y., Lucas, D., 2010, A literature review on mechanisms and models for the

coalescence process of fluid particles, Chem. Eng. Sci. 65, 2851-2864.

[1.84] Bertodano, M. L., Lahey Jr, R. T. and Jones Jr, O. C., 1994, Development of a

k-ε model for bubbly two-phase flow. J. Fluids Engineering 116, 128-134.

[1.85] Lucas, D., Tomiyama, A., 2011, On the lateral lift force in poly-dispersed

bubbly flows, Int. J. Multiphase Flow 37, 1178-1190.

37

Chapter 2

Experimental databases of upward turbulent bubbly

flow

2.1 Introduction

Before evaluating the existed models of the turbulent bubbly flow in the large

noncircular ducts, a comprehensive database is necessary to be collected. Table 2.1

summarized the database information according to its duct geometry, which will be used

in the present dissertation. Sun’s experimental result in the large noncircular ducts

[2.10] will be chosen as the main database in the following chapters. In the following,

the experimental database will be reviewed and collected including the experimental

techniques and apparatus, the flow conditions and the experimental results.

2.2 Experimental techniques

To measure the velocity and turbulence fields, nowadays several experimental

techniques have been developed and could be classified into contact type and

contactless type according to whether inserting the probes into the flow or not. The

contact type techniques insert the probes into the flow and thus there may exist some

effects of the inserted probes on the flow. Therefore, nowadays for the single-phase

turbulent flow measurement, the contactless measurement techniques such as laser

Doppler velocimetry (LDV), particle tracking velocimetry (PTV) and the particle image

velocimetry (PIV) played more and more important role in the understanding and

modeling of the single-phase turbulence. Generally, the above contactless techniques are

based on the optical techniques and require the transparent fluids property. However,

unlike the single-phase turbulent flow, due to the low visibility the above optical

38

measurement techniques failed for the turbulent bubbly flow except for the cases with

the few bubbles and in the near-wall region. The velocities and turbulence in the

turbulent bubbly flow were mainly measured by the contact type technique such as the

hot-film probe, based on which the velocity and the turbulent kinetic energy could be

measured at single point. However, it is notable that for the flow with large void fraction

the hot film is easy to be burn out due to the inefficient cooling by gas phase. These

phenomena also limit the application of hot film to the turbulent bubbly flow

measurements. Nonetheless, it has provided us many useful databases to understand the

physics and develop the model of turbulent bubbly flow.

As for the measurement of the bubbles’ behaviors, generally the flow

visualization is a useful way which also requires high visibility of the fluid properties

and thus could only be adopted in few bubbles cases. At present, for the turbulent

bubbly flow with many bubbles, the bubbles’ behaviors are mainly measured by probes

such as optical-sensor probes and conductive-sensor probes, based on which the bubble

number, concentration and the bubble geometry could be measured at the single point.

Fig. 2.1 summarizes the applications of the current experimental techniques for the

turbulent bubbly flow measurement.

2.3 Experimental apparatus

The experimental apparatus for the upward turbulent bubbly flow generally

contains five systems including the liquid phase circulation loop, the gas phase

supplying system, the inlet two phase mixing or bubble generation system, the phase

separation system and the test section with the measurement systems. Fig. 2.2 shows a

simplified schematic diagram of the experimental apparatus.

For the single phase flow, when the flows are introduced to the ducts there exist

two stages including the developing region with the entrance effects and the developed

region far from the inlet and without the entrance effects. In order to establish and

validate the models, the universal databases are necessary, which requires the

39

experimental measurements to be carried out in the fully developed region or

developing region with accurate control of the inlet flow. However, unlike the single

phase flow where the fully developed flow status could be reached with the long enough

pipes, for the turbulent bubbly flow the following problems make it difficult to establish

the universal database for the model development.

(a) Inlet flow control: not only the flow velocities, but also the distributions of

the void fraction and the bubble diameters should be considered.

(b) Bubble coalescence and breakup: they could change the bubbles properties

but their measurements and mechanisms are still in an initial stage.

(c) Bubble expansion: the bubbles properties could be changed for the high and

long ducts so that the fully-developed turbulent bubbly flow is difficult to

reach.

In summary, for the upward turbulent bubbly flow, after it is introduced to the

ducts, the flow includes the developing of the liquid phase flow field and the dispersed

bubbles distributions or flow regimes which couples together and determines the final

flow characteristics. It is more complicated for the flow to reach the fully-developed

status for both the liquid phase flow field and the dispersed bubbles distributions.

During the establishment and validation the turbulent bubbly flow models, the

developing status is necessary to be examined with the carefully controlled and

measured inlet parameters such as the distributions of void fractions, the bubble

diameters, and the velocities of liquid phase and bubbles. Table 2.1 shows the locations

of the experimental measurements in the current used databases which are still very

limited to understand the flow developing process.

40

Table 2.1 Experimental databases in different duct geometries. B: Base; H: Height

Duct shape Experimental

database

Duct size

DH (mm)

Experimental

techniques

Measured locations

z/ DH Measurements

Small

Circular Pipe

Serizawa et al. [2.1-2.2] 60 Dual-senor Conductive

& Hot-film probes 10, 20, 30

α, vb, fb

W, wrms

Wang et al. [2.3] 57.15 Hot-film probe 35 α, W, wrms, vrms, <vw>

Liu et al. [2.4-2.5] 38

57.2

Dual-senor Conductive

& Hot-film probes

36

30, 60, 90, 120

α, vb, fb, Db,

W, wrms, vrms, <vw>

Hibiki et al. [2.6] 25.4

50.8

Dual-senor Conductive

& Hot-film probes

12, 65, 125

6, 30.3, 53.5

α, vb, Db, ai

W, wrms

Small

Triangle Duct Bertodano et al. [2.7]

B: 50

H: 100 Hot-film probe 73 α, W, wrms, vrms, <vw>

Large

Circular pipe

Shawkat et al. [2.8] 200 Dual-senor optical &

Hot-film probes 42

α, vb, fb, Db,

W, wrms, vrms, <vw>

Sun et al. [2.9] 100 Multi-senor Optical &

Hot-film probe 3, 18, 33 α, vb, Db, ai

Large

Square Duct Sun [2.10] 136×136

Multi-senor Optical &

Hot-film probe 4.5, 10.1, 16

α, vb, fb, Db, ai

W, wrms, vrms, <vw>

41

Fig. 2.1 (a) Experimental techniques for the flow measurements and (b) its

applications in the turbulent bubbly flow.

42

Fig. 2.2 Simplified schematic diagram of the experimental apparatus.

2.4 Experimental results

Firstly, the flow conditions of the collected databases were shown in Tables 2.2

based on the area-averaged superficial liquid velocity Jl and gas velocity Jg for different

geometries. Meanwhile, the range of the flow regime is also shown.

For simplicity, only several representative sets of experimental results were

shown here in Figs. 2.3–2.11 including the axial liquid phase velocities Wl, void fraction

α, the turbulent intensities k or turbulent components <uu> or <vv> and the measured

bubble diameter DB. For the other sets used in the dissertation, it could be found in the

corresponding references.

As shown from Figs. 2.3–2.11 and mentioned in Chapter 1, for the upward

43

turbulent bubbly flow in the small circular pipes, the pronounced wall peak of the void

fraction could be observed for low area-averaged void fraction flow conditions. With

the wall peak of void fraction, the average axial liquid velocity (Wl) and turbulence

profiles are more uniform in the core region. For the turbulent bubbly in the large

circular pipes, the void fraction profile tends to be flatter as compared to the turbulent

bubbly flow in the small circular pipes. The wall-peak of the void fraction could only be

observed for the cases with extremely low averaged void fraction. Generally, in the both

the small and large circular pipes, comparing to the upward single-phase flow the

steeper ∩-shape with flatter velocity profile in the core region could be observed with

the wall peak of the void fraction. However, in the large square duct, not only the

pronounced wall and corner peaks of the void fraction, but also the significant peaks of

Wl could be observed near the corner and wall which form like the typical M-shaped

velocity distribution between two parallel walls and two diagonal corners. Hence, here

the apparent M-shaped Wl profile is regarded as one of typical characteristics of the

upward turbulent bubbly flow in this large noncircular duct. The detailed mechanism

will be considered in the Chapter 5.

2.5 Summary

In this chapter, the experimental databases used in the present dissertation were

collected including the flow conditions and the representative experimental results. In

the following chapters, the numerical evaluation and further analysis will be carried out

based on these experimental databases. It is notable that, in this chapter, generally two

sets of the experimental results were shown for simplicity. The more detailed database

could be found in the corresponding references.

44

Table 2.2 Experimental flow conditions in different duct geometries. B: Bubbly flow; T: Transition regime.

Duct shape Experimental

database

Flow conditions

Flow regime

<Jl> <Jg> , χ (For Serizawa [2.1-2.2])

Small

Circular Pipe

Serizawa et al. [2.1-2.2]

0.74

0.88

1.03

0, 0.0119, 0.0240, 0.0358

0, 0.0099, 0.0198, 0.03

0, 0.0085, 0.0170, 0.0258, 0.0341

B

Wang et al. [2.3] 0.43 0, 0.1, 0.27, 0.4 B to T

Liu et al. [2.4-2.5]

0.376

0.535

0.753

0.0, 0.18

0.18

0, 0.18

B to T

Hibiki et al. [2.6] 0.491

0.986

0, 0.0275, 0.0556, 0.129, 0.190

0, 0.0473, 0.0113, 0.242, 0.321 B to T

Small Triangle

Duct Bertodano et al. [2.7] 1.0 0.1 B

Large

Circular Pipe

Shawkat et al. [2.8]

0.45

0.58

0.68

0.015, 0.03,0.05, 0.065, 0.085, 0.1

0.015, 0.03, 0.065, 0.085, 0.1, 0.18

0.015, 0.03, 0.065, 0.085, 0.1, 0.18

B to T

Sun et al. [2.9] 1.021 0.1 B

Large square

duct Sun et al. [2.10]

0.50 0.0, 0.068, 0.090, 0.135 B to T

0.75 0.0, 0.068,0.090, 0.135, 0.180 B to T

1.00 0.0, 0.090, 0.135, 0.180, 0.225 B to T

1.25 0.0, 0.135, 0.180, 0.225, 0.270 B to T

45

Fig. 2.3 Lateral distributions of axial liquid phase velocities Wl and void fraction α in

the small circular pipe with DH=60mm under (a) Jl =0.74 m/s and (b) Jl =1.03 m/s based

on Serizawa’s experimental database [2.1].


pipe with DH=60mm under (a) Jl =0.74m/s and (b) Jl =1.03m/s based on Serizawa’s


0 0.2 0.4 0.6 0.8 10.5

1

1.5

r/R

Wl

Jl=0.74m/s

X %

0.000

0.0119

0.024

0.0358

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

r/R

Wl

J

l=1.03m/s

X %

0.000

0.0085

0.0170

0.0258

0.0341

0.0427

0.0511

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

r/R

<w

w>

Jl=0.74m/s

X %

0.000

0.0119

0.024

0.0358

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

r/R

<w

w>

J

l=1.03m/sX %

0.000

0.0085

0.0170

0.0258

0.0341

0.0427

0.0511

(b) (a)

(a) (b)

46


the small circular pipe with DH=50.8mm under (a) Jl =0.491m/s and (b) Jl =0.986m/s

based on Hibiki’s experimental database [2.6].


pipe with DH=50.8mm under (a) Jl =0.491m/s and (b) Jl =0.986m/s based on Hibiki’s


0 0.2 0.4 0.6 0.8 10.4

0.6

0.8

1

r/R

Wl

Jl=0.491/s

Jg

0.0275

0.0556

0.129

0.190

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

r/R

Wl

J

l=0.986/s

Jg

0.0473

0.113

0.242

0.321

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

r/R

<w

w>

Jl=0.491/s

Jg

0.000

0.0275

0.0556

0.129

0.190

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

r/R

<w

w>

Jl=0.986/s

Jg

0.000

0.0473

0.113

0.242

0.321

(b) (a)

(b) (a)

47

Fig. 2.7 Lateral distributions of (a) axial turbulent kinetic energy <ww> and lateral turbulent kinetic energy <uu> and (b) void fraction α

in the small circular pipe with DH=60mm based on Liu’s experimental database [2.3-2.4].

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

<w

w>

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

r/R



Jl J

g

0.376 0.000

0.753 0.000

0.376 0.018

0.535 0.018

0.753 0.018

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

r/R

0.376 0.018

0.535 0.018

0.753 0.018

(a) (b) Jl Jg

48


the large circular pipe with DH=200mm under (a) Jl =0.45m/s and (b) Jl =0.68m/s based

on Shakwat’s experimental database [2.7].


turbulent kinetic energy <uu> in the large circular pipe with DH=200mm under (a) Jl

=0.45m/s and (b) Jl =0.68m/s based on Shakwat’s experimental database [2.7].

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Wl

Jl=0.45m/s

Jg

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

r/R

0.0000.050

0.0850.100

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Wl

Jl=0.68m/s

Jg

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

r/R

0.0000.065

0.0850.100

0.180

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

<w

w>

Jl=0.45m/s

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

r/R



Jg

0.0000.050

0.0850.100

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

<w

w>

Jl=0.68m/s

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

r/R



Jg 0.000

0.0650.0850.100

0.180

(b) (a)

(b) (a)

49

Fig. 2.10 Lateral distributions of axial liquid phase velocities Wl and void fraction α in the large circular pipe with DH=136mm:

under Jl =0.75m/s (a) along the bisector line; (b) along the diagonal line; under Jl =1.00m/s (c) along the bisector line; (b) along the

diagonal line based on Sun’s experimental database [2.10].

0 0.2 0.4 0.6 0.8 10.5

1

1.5

Wl

Jl=0.75m/s Bisector

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

2x/DH

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10.5

1

1.5

Wl

Jl=0.75m/s Diagonal

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

2x/DH

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

Wl

Jl=1.0m/s Bisector

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

2x/DH

0.0000.0900.135

0.1800.225

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4

1.6

Wl

Jl=1.0m/s Diagonal

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

2x/DH

0.0000.0900.135

0.1800.225

(b) (a)

(d) (c)

Jg Jg

Jg

Jg

50

Fig. 2.11 Lateral distributions of axial turbulent kinetic energy <ww> and lateral turbulent kinetic energy <uu> in the large circular pipe

with DH=136mm under Jl =0.75m/s (a) along the bisector line; (b) along the diagonal line; under Jl =1.00m/s (c) along the bisector line;

(d) along the diagonal line based on Sun’s experimental database [2.10].

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

<w

w>

Jl=0.75m/s Bisector

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

2x/DH



0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

<w

w>

Jl=0.75m/s Diagonal

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

2x/DH



0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

<w

w>

Jl=1.0m/s Bisector

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

2x/DH



Jl=1.0m/s Diagonal

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

2x/DH



0.0000.0900.135

0.1800.225

(b) (a)

(d) (c)

Jg

Jg

Jg

Jg

51

Reference



[2.2] Serizawa, A., Kataoka, I., Michiyoshi, I., 1975, Turbulence structure of air-water

bubbly flow—II. local properties, Int. J. Multiphase Flow 2, 235–246.

[2.3] Wang, S., Lee, S., Jones, O., Lahey, Jr R. T., 1987, 3-D turbulence structure and


Flow 13, 327–343.


pipe—I. bubble void fraction, bubble velocity and bubble size distribution, Int. J.

Heat Mass Transfer 36, 1061–1072.



Transfer 36, 1049–1060.

[2.6] Hibiki, T., Ishii, M., Z. Xiao, 2001, Axial interfacial area transport of vertical

bubbly flows, Int. J. Heat Mass Transfer 44, 1869-1888.

[2.7] Bertodano, de M. L., 1992, Turbulent bubbly two-phase flow in a triangular duct.

Doctoral Thesis, Resselaer Polytechnic Institute.








52

53

Chapter 3

Numerical evaluation of two-fluid model of turbulent

bubbly flow in large square duct

3.1 Introduction

As reviewed in Chapter 1, numerical efforts on turbulence modeling of bubbly

flow have been made based on an assumption of isotropic turbulence, such as k-ε model.

Moreover, the numerical evaluations of the previous models were mainly carried out

based on the experimental database in the small circular and noncircular ducts. However,

for the single-phase turbulent flow in the noncircular ducts, the secondary flow exists in

the cross-section due to the anisotropic turbulence, which has strong effects on the flow

characteristics such as distorting the local mean velocity distribution. Therefore, for the

turbulent bubbly flow in the noncircular ducts, the interaction between the secondary

flow and the bubbles could exist. In addition, the turbulent bubbly flow in the large duct

shows different flow characteristics from that in the small ducts. Thus, the models

mentioned above need to be evaluated for the turbulent bubbly flow in the large and

noncircular ducts. In this chapter, the numerical evaluation of the existed two-fluid

model will be carried out for the turbulent bubbly flow in the large square duct based on

the recently established database by Sun [3.1].

3.2 Numerical methodology

3.2.1 Problem description

The numerical evaluation here is carried out by mimicking previous simulation

54

of the turbulent bubbly flow in the circular pipe by changing the shape of the duct

geometry as shown in Fig. 3.1. Similar to that in [3.1], the duct size is chosen to be

0.136m×0.136m×3m. The bubble size DB is chosen according to the different

experimental conditions.

Fig. 3.1 Schematic diagram of the numerical problem.

3.2.2 Numerical methods

3.2.2.1 Governing equations

Considering the incompressible liquid and gas phase without phase change, the

governing equations of the turbulent bubbly flow based on Eulerian-Eulerian two-fluid

model [3.2] could be given as follows including the mass and momentum conservation.

/ 0gg gt V , (3.1)

55

/ 0ll lt V , (3.2)

Re

/ -

T

g g g g gg g g g g g g

g g g g g ig

t p

V V V V V

τ g M

, (3.3)

Re

/ -

T

l l l l ll l l l l l l

l l l l l il

t p

V V V V V

τ g M

, (3.4)

where the definitions of the variables are the same as that defined in Chapter 1. In

addition, the void fraction satisfies 1 gl ; the phase interaction through the

interfaces satisfies 0il ig M M ; the Reynolds stress term Reg g g τ in the gas phase

could be neglected due to lg .

3.2.2.2 Constitutive relations

The following constitutive relations were adopted for the interfacial and

turbulence terms to close the above equations:

For the interfacial term Mil, the phase interaction was considered by a drag force

MD [3.2], a virtual mass force MVm [3.3], a lift force ML [3.4], a turbulent dispersion

force MT [3.5], a wall force Mw [3.6], and a pressure force Mp[3.7].

il ig D VM L T W p M M M M M M M M . (3.5)

The following constitutive relations were adopted for each force:

3

- -4

g l g lD D l g

B

CD

M V V V V , (3.6)

-g l lL g L lC M V V V , (3.7)

Dg lVM g l VM Dt

C M V V , (3.8)

l

l

g

gtDTDT CC

-M , (3.9)

56

2

W -g lW g l wC M V V n , (3.10)

2

p -g lp g l iC M V V n , (3.11)

where CD, CL , CVM, CTD, CW , and Cp are the coefficients for each force and were

chosen as follows:

0.68724 72 8 Eomax min 1 0.15Re , ,

Re Re 3 Eo 4DC

;

min 0.288tanh 0.121Re , Eo For Eo 4

Eo For 4 Eo 10

0.29Eo For 10

m

L

f

C f

Eo

;

2/1VMC ,

wB

WyD

C05.001.0

,0max ; 1.0TDC ; 0.1pC

For the turbulence term τRel in Eq. (3.4), the η-ε model proposed by Lopez

Bertodano et al. [3.7] was adopted, which considers the turbulence in the liquid phase

τRel separately including the shear-induced turbulence τs and bubble induced turbulence

τB.

BS τττ Re . (3.12)

The shear-induced turbulence τs and bubble-induced turbulence τB are

respectively given by

SSl

T

llleffS kAVVτ 3

2, , (3.13)

BBlB kAτ 3

2 , (3.14)

where AS and AB are the anisotropy matrices. leff , is the effective eddy viscosity and

was chosen based on the k-ε model and Sato model [3.8] as

lgbglbSltbltleff uuDCkC /2

,, , (3.15)

where Cμl and Cμb are the model constants and chosen to 0.09 and 0.6 respectively.

57

To close Eq. (3.13) and (3.15), the shear-induced turbulent kinetic energy kS was

considered by the well-known k-ε model for the single-phase turbulent flow as

-Sll S l t S l

kk k P

t

V , (3.16)

2

1 2-tll l l

S S

PC C

t k k

V , (3.17)

where ,

T

l l leff lP

V V V ; ζε, Cε1 and Cε2 are model constants and are the

same as that in the single-phase k-ε model with ζε=1.3, Cε1=1.44, and Cε1=1.92,

respectively.

Considering the bubble-induced turbulent kinetic energy kB as a scalar, the

transport equations could be given as

-B l Tll B l t B Ba B

B

k Vk k k k

t D

V (3.18)

where VT is the terminal velocity of a single bubble and kBa is the asymptotic value of

the bubble-induced kinetic energy 21

2g lBa g VMk C V V and 2VMC

.

3.2.2.3 Boundary conditions

The boundary conditions especially at the wall are essential for the numerical

simulation and given as follows: In the inlet, the uniform boundary conditions were

adopted including uniform liquid and gas superficial velocities Jl and Jg and turbulence.

In the outlet, the natural flow conditions were adopted by giving the outlet pressure as

the atmosphere. At the wall, for the liquid phase, similar to the single-phase turbulent

flow, the wall function was used for the velocity and turbulence [3.9]. For the gas phase,

the void fraction at the wall was set to be 0Wallg indicating the bubble can’t enter

the wall.

58

3.2.2.4 Numerical procedures

In the present dissertation, the governing equations were numerically solved by

finite differential method, which could be briefly summarized as follows: Firstly, the

discretization of the above equations could be rearranged as

11 nn n n n n n

gg g g g g g A G V A , (3.19)

11 nn n n n n n

ll l l l l l A G V A , (3.20)

1 1n n nn n n n n n n

g g gVg g Vg Vg g Vgp A V H G V A V , (3.21)

1 1 n nn n n n n n n nl lVl l l Vl Vl l Vlp A V H G V A V , (3.22)

1n n n n n n n n

ks l S ks S ks l Sk k k A G A , (3.23)

1n n n n n n n n

s l S s S ks l S A G A , (3.24)

1n n n n n n n n

kB l B kB B kB l Bk k k A G A , (3.25)

where A, H and G are matrices related with the discretization of the time-derivative,

pressure term and the other terms including convection, diffusion, interfacial interaction,

turbulence and source terms.

Then, the projection method was used to solve the pressure filed p and update the

velocity field V as follows:

* /n nn n n ng gg Vg g Vg Vg V G V A V A , (3.26)

* /n nn n n n nl ll Vl l Vl Vl l V G V A V A , (3.27)

1 * 1 /

n n n n ng g Vg Vg gp V V H A , (3.28)

1 * 1 /

n n n n nl l Vl Vl lp V V H A , (3.29)

Substituting Eqs. (3.28) and (3.29) to the summation of Eqs. (3.19) and (3.20)

with the constraint of αg+αl=1 and rearranging, the pressure filed could be solved by

59

1 * *n n n n

l gp B C V D V , (3.30)

where B, C and D are matrices related with grid size.

With the new pressure field, the fields of velocities in liquid and gas phases, void

fraction and turbulence could be updated. Fig. 3.2 shows the flow chart which was used

to solve the governing equations.

3.3 Results and discussion

During the numerical calculation, the inlet flow rates and bubble size were set to

be similar to that in the experiments [3.1]. Figs 3.4 show the comparisons of the void

fraction, liquid phase velocities and turbulence between the experimental measurements

and predictions under Jl =0.75m/s and Jg =0.09m/s. For simplicity, the comparisons of

the other cases were not shown here since they showed the similar tendency. It is

obtained that the current k-ε model could predict the corner peak phenomenon of void

fraction and local liquid velocity. However, the predicted values of the peak void

fraction and velocity showed much smaller as compared with experiments, especially,

the turbulence fields.

The following factors are considered to be responsible for the above

insufficient prediction. (1) As compared with the experiments, the existing model

predicted the almost constant relative velocity Wr around 0.23m/s. However, in the

experiments, the relative velocity Wr was related with the void fraction and shows

smaller than the prediction. (2) Based on the k-ε model, the secondary flow effect was

neglected, which could affect the bubbles motion and push the bubbles to the corner. In

addition, unlike the circular pipe with a homogeneous boundary, the small bubbles

could be trapped in the corner region because of the non-homogeneous boundary. (3)

The model constants were mainly proposed and validated based on the experiments in

60

the small ducts. It might not be suitable in the large ducts however so far our current

understandings of the physical process therein are limited. Therefore, to improve the

model for the turbulent bubbly flow in the large noncircular ducts, firstly the

understanding of the physical process in the large noncircular ducts is urgently in need,

which is the target in the following four chapters.

3.4 Summary

In this Chapter, the numerical evaluations of the current two-fluid model of the

turbulent bubbly flow were carried out in the large square duct based on the recently

established database. It showed that the current model might not be suitable for the

prediction of the turbulent bubbly flow in the large noncircular ducts because of the

different physical processes therein as compared to that in the small circular pipes. In

order to improve the model therein, the further detailed analysis of the physical process

of the turbulent bubbly flow in the large noncircular ducts will be suggested such as the

liquid phase velocity peak near the wall, the significant corner peak of the void fraction,

and strong turbulence.

61

Fig. 3.2 Flow charts of numerical calculation.

62

Fig. 3.3 Comparisons of the experimental measurements and numerical model under Jl =0.75m/s and Jg =0.09m/s:

for the liquid phase velocity (a) along the bisector line and (b) along diagonal line; for the void fraction (c) along the bisector line and

(d) along diagonal line; for turbulence (e) along the bisector line and (f) along diagonal line.

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

2x/DH

Wl

Experimental data

Existing model

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

2x/DH

Experimental data

Existing model

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

2x/DH

t k

Experimental data

Existing model

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

2x/DH

Wl

Experimental data

Existing model

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

2x/DH

Experimental data

Existing model

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

2x/DH

t k

Experimental data

Existing model

(a)

(b)

(c)

(d)

(e)

(f)

63

Reference



[3.2] Ishii, M., Hibiki, T., 2011, Thermo-fluid dynamics of two-phase flow. 2nd ed.

New York: Springer.





[3.5] Burns, D., Frank, T., Hamill, I., Shi, J. M., 2004, The Favre averaged drag

model for turbulent dispersion in Eulerian multi-phase flow, Proc. 5th

International Conference on Multiphase Flow, Yokohama, Japan, 392.

[3.6] Antal, S.P., Lahey, Jr R. T., Flaherty, J. E., 1991, Analysis of phase distribution


309-326.







64

65

Chapter 4

Developing process of turbulent bubbly flow

4.1 Introduction

Several researches have discussed the developing of the turbulent bubbly flow

but mainly focused on the general flow pattern transition from the bubbly flow to the

slug flow [4.1-4.3]. Few researches were focused on the detailed developing process

based on the bubble motion. Consequently, there does not exist any detailed model

about the formation of the bubble layer, the velocity and the turbulence distribution

during the developing process. Similar to the single-phase turbulent flow, in order to

develop the developing models for the turbulent bubbly flow, the experimental

databases are essentially required with the well-controlled inlet flow conditions

including not only velocity distributions but also the bubble distribution. However, for

the circular pipes the experimental databases could not satisfy the requirements which

were either with complicated inlet flow conditions [4.4-4.5] or measured far from the

inlet [4.6]. In Sun’s experiments [4.7], the experimental measurements were carried out

at different locations during the flow developing with well-controlled inlet flow

conditions, which provides us the possibility to consider the developing process. In this

chapter, the developing process in the turbulent bubbly flow will be discussed based on

the database in [4.7].

4.2 Experimental observations

Firstly, based on Sun [4.7] Figs. 4.1-4.4 show the experimental results of the

66

liquid phase velocities and the corresponding void fractions at different locations for the

different flow conditions summarized in Table 4.1. From the inlet to the measured

location, generally two stages could be observed during the developing including the

formation of the bubble layer near the wall and the developing the liquid phase with this

bubble layer the image of which was shown in Fig. 4.5. Moreover, the developing

processes for the small void and large void fraction cases in the transition region

showed quite different characteristics. For the cases with large void fraction, after the

formation of the bubble layer, the further rearrangement of the bubbles could be

observed due to the coalescence of the bubbles occur near the wall, which made the

bubble lateral migration from the wall to the center. This process dominates the second

stage of the developing process and was more complicated. In this chapter, we will

discuss the developing processes during these two stages especially for the small void

fraction. The following topics will be focused on. In the first stage, the formation of the

bubble layer will be discussed by considering how long it needs to form the developed

bubble layer. In the second stage, the length required to lift up the liquid phase velocity

will be discussed.

Table 4.1 Experimental conditions for the developing process.

Jl (m/s) 0.5 0.75 1.0 1.25

Jg (m/s) 0.09 0.135 0.09 0.135 0.18 0.09 0.135 0.18 0.135 0.18 0.225

67

Fig. 4.1 Liquid phase velocities Wl and the corresponding void fractions α at z=4.5 DH, 10 DH and 16DH for Jl=0.5m/s:

with Jg=0.09m/s along (a) the bisector line; (b) the diagonal line; with Jg=0.135m/s along (c) the bisector line; (d) the diagonal line [4.7].

0

0.5

1

Wl

Jl=0.5m/s J

g=0.09m/s Bisector

0 20 40 60

0

0.2

0.4

Distance from the wall x mm

z=4.5DH

z=10DH

z=16DH

0

0.5

1

Wl

Jl=0.5m/s J

g=0.09m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4

Distance from the corner x mm

z=4.5DH

z=10DH

z=16DH

0

0.5

1

Wl

Jl=0.5m/s J

g=0.135m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

Wl

Jl=0.5m/s J

g=0.135m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

(a) (b)

(c) (d)

68


with Jg=0.09m/s along (a) the bisector line; (b) the diagonal line; with Jg=0.135m/s along (c) the bisector line; (d) the diagonal line;

with Jg=0.18m/s along (e) the bisector line; (f) the diagonal line [4.7].

0

0.5

1W

l

Jl=0.75m/s J

g=0.09m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

Wl

Jl=0.75m/s J

g=0.135m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl

Jl=0.75m/s J

g=0.18m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl

Jl=0.75m/s J

g=0.09m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl

Jl=0.75m/s J

g=0.135m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl

Jl=0.75m/s J

g=0.18m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

(a)

(b)

(c)

(d)

(e)

(f)

69

Fig. 4.3 Liquid phase velocities Wl and the corresponding void fractions α at z=4.5 DH, 10 DH and 16DH for Jl=1.0 m/s:

With Jg=0.09m/s along (a) the bisector line; (b) the diagonal line; with Jg=0.135m/s along (c) the bisector line; (d) the diagonal line;


0

0.5

1

1.5

Wl

Jl=1.0m/s J

g=0.09m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl J

l=1.0m/s J

g=0.135m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

1

2

Wl

Jl=1.0m/s J

g=0.18m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl

Jl=1.0m/s J

g=0.09m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

0.5

1

1.5

Wl J

l=1.0m/s J

g=0.135m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

0

1

2

Wl

Jl=1.0m/s J

g=0.18m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

(a)

(b)

(c)

(d)

(e)

(f)

70


with Jg=0.135m/s along (a) the bisector line; (b) the diagonal line; with Jg=0.18m/s along (c) the bisector line; (d) the diagonal line;


1

1.5

2W

l

Jl=1.25m/s J

g=0.135m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

1

1.5

2

Wl

Jl=1.25m/s J

g=0.18m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

1

1.5

2

Wl

Jl=1.25m/s J

g=0.225m/s Bisector

0 20 40 60

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

1

1.5

2

Wl

Jl=1.25m/s J

g=0.135m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

1

1.5

2

Wl

Jl=1.25m/s J

g=0.18m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

1

1.5

2

Wl

Jl=1.25m/s J

g=0.225m/s Diagonal

0 20 40 60 80 100

0

0.2

0.4


z=4.5DH

z=10DH

z=16DH

(a)

(b)

(c)

(d)

(e)

(f)

71

Fig. 4.5 Schematic diagram of the developing process of the upward turbulent bubbly

flow in the large square duct.

4.3 Modeling of developing process

4.3.1 Formation of bubble layer in the first stage

In order to determine the length required for the formation of the bubble layer

LδB, the following two questions are necessary to be considered, i.e., why the lateral

migration of the bubbles occurs and when the stable bubble layer could be formed. As

mentioned in Chapter 1, the bubble lateral accumulation in the duct flow results from

the lift force because of the liquid phase velocity gradient in the lateral direction which

is dominated by the viscosity and the wall effect. Therefore, the process of the bubble

layer formation could be modeled as follows. Given the bubbly flow with perfect inlet

boundary conditions which are the uniform distributions of the bubbles liquid phase

velocity and turbulence, two processes coupled together during the flow developing

process including the developing of the liquid boundary layer and the bubble layer.

Then, when the bubble layer could be formed? The lateral force balance exerted on the

bubbles as shown in Fig. 4.6 dominates the bubbles lateral migration and the formation

of the bubble layer. In addition to the wall lubrication force [4.8], the turbulence

developing near the wall due to the shear and bubbles could induce the lateral turbulent

dispersion force [4.9] toward the center. Moreover, the coalescence of the bubbles due

72

to accumulation of the bubbles near the wall will also occur, which increased the bubble

size and decreased the lift force. When the lateral forces reached the balanced status, the

bubble layer could be formed. Herein, the flow with the small void fraction without the

bubble coalescence effect will be only considered. Therefore, the developing process of

the bubble layer is dominated by the developing of the liquid phase boundary layer

including the liquid phase velocity which generated the lateral lift force and the

turbulence near the wall which generated the turbulent dispersion force balanced with

the lift force. The length required to form the bubble layer LδB then depends on the

developing process of the liquid phase boundary layer.

Fig. 4.6 Schematic diagram of the formation of the bubble layer due to the lift force

exerted on the bubbles and the developing of the boundary layer in the liquid phase.

The following method is used to determine the developing length of the bubble

layer LδB. Firstly, the following assumptions are considered: (1) The uniform velocity

and bubble distribution in the inlet; (2) The little bubble effect on the liquid flow

developing during the lateral accumulation of the bubbles; (3) Neglecting the turbulent

73

dispersion force; (4) During the formation of boundary layer, only the neighbor bubble

could enter this layer immediately and other bubbles will also be mixed immediately.

With the above assumptions, the formation of the bubble layer with the thickness δB at

least requires the existence of the liquid phase boundary layer with the thickness δl = δB,

i.e., the length from the inlet for the bubble layer formation LB could be reflected by the

length from the inlet for liquid-phase boundary layer formation with the thickness Lδl.

Fig. 4.7 Comparison between the thickness of the boundary layer δl and the bubble layer

δB at the measurement location z=4.5DH.

Then, the bubble effect on the flow developing and the turbulent dispersion

force effect will be considered. As compared to the single-phase flow, the liquid phase

velocity in the boundary layer will be lifted up by the bubble accumulation and

buoyancy effect therein, which will lead to reduction of the lift force. The turbulent

dispersion force exerted on the bubbles will resist the bubbles moving into the boundary

layer. Therefore, the thickness of the bubble layer δB will be smaller than that of the

boundary layer δl, or the developing length LB will be shortened by these two effects,

0 0.005 0.01 0.015 0.02 0.025 0.030

0.5

1

1.5

2

Turbulent boundary layer thickness m

Dis

tan

ce f

rom

th

e u

nif

orm

in

let

z m

/z = 0.37Rez

-1/5

Wm

=0.50 m/s

Wm

=0.75 m/s

Wm

=1.00 m/s

Wm

=1.25 m/s

Thickness of bubble layer

z=4.5DH

z=3.3DH

z=2.8DH

Bubble layer formed

74

i.e., LB > Lδl. Here, the developing process of the liquid phase boundary layer will be

considered based on the single-phase turbulent boundary layer theory [4.10]:

1/5/ 0.371Rel zz , where Re /z Vz . Fig. 4.7 shows the thickness of the boundary

layer δl during the developing and the bubble layer δB at the measurement location

z=4.5DH. It could be observed that based on the above assumptions δl > δB at the

measurement location z=4.5DH, indicating the bubble layer could be already formed

before entering the measurement location.

4.3.2 Liquid phase velocity lifting in the second stage

Due to the formation of the bubble layer near the wall which made the fluid

lighter than that in the core region, the liquid phase velocity near the wall will be lifted

up by the bubbles. Fig. 4.8 shows the schematic diagram of liquid phase velocity lifting

up in the second stages as shown in Fig. 4.5. In the part, the length required to lift up the

liquid phase velocity Lp will be discussed based on the dimensional analysis as follows.

Fig. 4.8 Schematic diagram of liquid phase velocity lifting up in the second stage.

After forming the bubble layer near the wall, the liquid phase therein will be

accelerated due to the buoyancy effect. Here, considering the force balance on the liquid

75

phase, the acceleration rate al could be estimated as

1/gal . (4.1)

For different mean liquid phase velocity W, the time tl required to lift the velocity near

the wall from h1×W to h2×W could be estimated as

2 1 1 /lt h h W g . (4.2)

Then the length Lp required to lift the velocity near the wall from 0 to h×W could be

estimated as considering the acceleration

2 2 2

2 1 1 / 2pL h h W g . (4.3)

The dimensionless parameter A is defined as follows which shows the

developing status of the liquid phase velocity and ratio between the measured location

zp and Lp:

2 2 2

2 12 / 1p

p

p

zA z g h h W

L

, (4.4)

Fig. 4.7 shows the ratio A based on the experimental results corresponding to the

cases as shown in Table 4.1. For these cases, the measurements were carried out for

three locations zp =4.5DH, 10DH and 16DH. There might exist a critical value AC,

according to which, the velocity developing status could be judged by

For not lifted up

For lifted up

C

C

A A

A A

. (4.5)

According to the experimental results shown in Fig. 4.2, it could be obtained

that for the cases with Jl = 0.75m/s and Jl = 0.09m/s, the liquid phase velocities near the

wall shows still flat at 4.5DH and then were lifted up at z=10DH and stable until 16DH.

Thus, the dimensionless parameter A for this case at z=10DH is chosen as a criterion AC.

76

Correspondingly, the following conjectures for the other cases could be obtained: For

the cases with Jl = 0.5m/s, the liquid phase velocities could be lifted up before 4.5DH.

For the cases with Jl = 1.0m/s, the liquid phase velocities could be lifted up around or

after 10DH. For example, it is around 10 DH for Jg = 0.18m/s, 16 DH for Jg = 0.135m/s,

but more than 16 DH for Jg = 0.09m/s. For the cases with Jl = 1.25m/s, except Jg =

0.225m/s, more than 16 DH is required to lift up the liquid phase velocities near the wall.

By comparing with Figs. 4.1 -4.4 the above conjectures could be well confirmed, which

indicates the dimensionless parameter A could be a proper criterion to judge the flow

developing status.

Fig. 4.9 Comparison of dimensionless parameter 2/ 1pA L g W for different

superficial liquid and gas phase velocities Jl and Jg and measurement locations.

0 0.5 1 1.5 2 2.50

5

10

15

20

Jl

A

z=4.5DH

z=10DH

z=16DH

Ac=2.8

0.5 1 1.50

2

4

6

<Jl>

A

Jl

77

4.4 Summary and Discussion

In Section 4.2-4.3, the developing processes of the turbulent bubbly flow were

analyzed based on Sun’s experimental results [4.7]. Two stages were observed during

the flow developing which are forming the bubble layer and the velocity developing

based on the formed bubble layer. At the first stage of the bubble layer formation, the

length required to form the bubble layer was analyzed, which was found to be less than

that required to develop the liquid phase boundary layer with the similar thickness. At

the second stage of the velocity developing, the length required to lift up the liquid

phase velocity near the wall was analyzed, based on which the criterion to judge the

flow developing status was suggested.

However, so far the lack of the detailed experimental databases during the flow

developing process strongly limited us to verify the above analysis and to further

understand and model of this process. Many interesting questions and modeling are still

unable to be considered including:

(1) For the turbulent bubbly flow the thickness of the bubble layer δB was

observed to around 2δB/DH ≈0.2. And why the bubble layer was formed with

thickness? During the formation of the bubble layer, how do the distributions

of the phase, liquid velocity and turbulence evolve?

(2) At second stage where the velocity develops with bubble layer, how much

does it need to develop the velocity such as lift up the local liquid velocity?

After lifting up the liquid phase velocity near the wall, how are the bubbles

and liquid phase velocity rearranged?

Therefore, in order to understand the developing process of the turbulent

bubbly flow, the boundary layer flow type with careful design and detailed

measurements are suggested.

78

Reference

[4-1] Kaichiro, M., Ishii, M., 1984, Flow regime transition criteria for upward

two-phase flow in vertical tubes, Int. J. Heat Mass Transfer 27, 723–737.

[4-2] Schlegel, J.P., Sawant, P., Paranjape, S., Ozar, B., Hibiki, T., Ishii, M., 2009,

Void fraction and flow regime in adiabatic upward two-phase flow in large

diameter vertical pipes, Nucl. Eng. Des. 239, 2864–2874.

[4-3] Ohnuki, A., Akimoto, H., 2000, Experimental study on transition of flow pattern

and phase distribution in upward air–water two-phase flow along a large vertical

pipe, Int. J. Multiphase Flow. 26, 367–386.

[4-4] Serizawa, A., 1974, Fluid dynamics characteristics of two-phase flow, Doctor


[4-5] Wang, S., Lee, S., Jones, O., Lahey, Jr R. T., 1987, 3-D turbulence structure and


Flow 13, 327–343.

[4-6] Shawakat, M. E., Ching, C. Y., Shoukri, M., 2008, Bubble and liquid turbulence



[4-7] Sun, H. M., 2014, Study on upward air-water two-phase flow characteristics in a


[4-8] Antal, S.P., Lahey, Jr R. T., Flaherty, J. E., 1991, Analysis of phase distribution


309-326.

[4-9] Lahey, Jr R. T., , Bertodano, de M. L., Jones, O. C., 1993, Phase distribution in

complex geometry conduits, Nucl. Eng. Des. 141,177-201.

[4-10] Schlichting, H., Gersten, K., 2008, Boundary layer theory, 8th

ed., Springer.

79

Chapter 5

Axial liquid-phase velocity of turbulent bubbly flow

5.1 Introduction

As shown in Fig. 2.9, for the turbulent bubbly flow in the large square duct [5.1],

not only the pronounced wall and corner peaks of void fraction, but also the significant

peaks of liquid phase velocities were observed near the corner and wall which form like

the typical M-shaped velocity distribution as shown in Fig. 5.1(a) between two parallel

walls, especially two diagonal corners. Generally, for the upward turbulent bubbly flow

in the small and large circular pipe, the steeper ∩-shape with flatter velocity profile in

the core region as compared with the upward single-phase flow was observed [5.2-5.6]

even with the significant wall peak of the void fraction. Therefore, the apparent

M-shaped Wl profile could be regarded as one of typical characteristics of the upward

turbulent bubbly flow in this large square duct where both the scale and the

non-homogeneous geometrical boundary effects are important. In this chapter, we firstly

discussed the mechanisms for formation of typical M-shaped Wl profile observed in the

previous upward turbulent bubbly flow experiments [5.1]. Here, three topics about the

formation of this typical velocity profile will be firstly discussed including the index of

flow type, the conditions for the formation of the M-shaped Wl profile and the peak

location of Wl. Based on that, the duct geometry effects on the void fraction profile and

wall shear stress will be considered. Then, the flow characteristics will be discussed in

two layers based on the normalization of obtained axial liquid phase velocity. Finally,

the numerical validation of the axial liquid phase velocity will be carried out.

80

Fig. 5.1 Schematic diagram of the type of the velocity profile of the upward flow:

(a) M-shape; (b) ∩-shape; (c) Cuspid-shape; (d) W-shape

5.2 Index of flow type

In this part, the index of the flow type as shown in Fig. 5.1 will be focused on in

order to consider the formation conditions of M-shaped Wl profile. Generally the

formation of different velocity profile in the upward turbulent bubbly flow could be

attributed to the counteraction between the wall and bubble effect which exert retarding

and pushing effect on the liquid flow, respectively.

To begin with, the wall effects on upward flow driven by pressure drop were

considered. Generally, the wall retards the main flow because of the nonslip condition at

the wall and fluid viscosity. For the single-phase upward flow with the constant density

the wall retarding effect is exerted on the whole area, so the ∩-shaped velocity profile

shown in Fig. 5.2(b) could be formed. However, for the upward turbulent bubbly flow,

the bubbles lateral migration due to the lift force result in the non-uniform density of the

fluid ρf =αlρl +αgρg. Hereafter, the symbols α and ρ are the void fraction and density,

and the subscripts ―l‖ and ―g‖ indicate the liquid and gas phase, respectively. The area

with wall retarding effect could be modified by the non-uniform density distribution.

To show the coupling interaction between the wall and non-uniform density

effect, the local net driving force F was evaluated. Considering the mixture of the gas

and liquid phases, the averaged local net driving force F directed to the upward in the

81

cross section could be calculated by

ggdzdPF ggll / , (5.1)

where dp/dz is the mean pressure gradient of the mixture and the main driving force

coming from the pump and compressor; g is gravity acceleration: -9.81m2/s. The gas

phase could be neglected since ρg << ρl. Here assuming fully-developed flow status and

neglecting the flow acceleration and secondary flow, the net driving force F is balanced

with the local net viscous force dηv/ds (ηv: local viscous force and s: coordinate direction,

x or y) from the neighbor fluid elements due to the molecular and turbulent eddy

viscosities.

Next, we consider the indications of physical process between the wall and

center by the sign of local net driving force F. If locally F > 0, it indicates the pressure

gradient is able to overcome the local gravity, and the flow could be pushed upward

driven by the pressure gradient. The local net viscous force from the neighbor fluid

element is downward and retards the local flow as a flow resistance. But if locally F < 0,

it indicates the pressure gradient is unable to overcome the local gravity. The fluids tend

to fall downward and keep the flow upward because the local net viscous force is

upward which pulls the flow as the additional driving force. In this case, the flow is

driven by the viscous shear force.

For the fully-developed turbulent flow, we assume the pressure gradient dp/dz to

be uniform among the cross sections. With the above indication, for the upward

single-phase flow driven by the pressure gradient, the net driving force F is uniform and

positive. Due to the wall, the local net viscous force always acts as the flow resistance

and the ∩-shaped velocity profile is formed. However, for the upward turbulent bubbly

flow, the net driving force F is non-uniform due to the non-uniform density among the

cross sections. According to the sign of F, the formation of different shape velocity

82

profile could be implied.

In [5.1], the mean pressure gradient dp/dz and αg distributions were measured

simultaneously and assuming dp/dz to be uniform among cross sections, the lateral

distributions of F were calculated and shown along the bisector and the diagonal lines,

respectively in Figs. 5.2-5.3. Table 5.1 shows the experimental data of the mean

pressure gradient dp/dz for different flow conditions at z=16DH based on the pressure

and height difference between z=15DH and z=19DH measured by the differential

pressure transmitters. Besides, the void fraction distribution at z=10.1DH was used since

the flow at z=16DH was developed based on void fraction distribution before entering

this part such as z=10.1DH. Besides, the void fraction distribution has already reached

the developed status at z=4.5DH as discussed in Section 4.2, so the choice of void

fraction distribution at different locations have no effect on the following analysis.

Table 5.1 Experimental data of dp/dz (kPa/m).

Jg (m/s)

Jl (m/s)

0.090 0.135 0.180 0.225 0.315

0.75 8.90 8.50 8.12 7.86 -

1.00 9.10 8.78 8.48 8.19 7.98

As shown in the above figures, the obvious sign changing of F at point xp near

the wall and corner could be observed for the turbulent bubbly flow in the large square

duct. Moreover, completely different behaviors were observed for the small,

intermediate and large void fraction cases. For the small void fraction case, the local net

driving force F is positive between xp and the wall, but negative between xp and the duct

center, indicating that the local net viscous force behaves as flow resistance between xp

83

and wall but driving force between the point xp and the duct center. Because of the wall

retarding effect, for the small void fraction cases the viscous force drags the flow near

the wall. After xp, the flow is driven by the local viscous force due to the high velocity

near the wall, indicating the wall retarding effect disappears or is blocked by the

bubbles accumulated between the wall and xp. It is notable that when F has a sign

changing point, the cuspid-shaped velocity profile shown in Fig. 5.1(c) also implying

the role of local net viscous force changes from the drag force to the driving force.

However, since there is no driving force in the duct center, this flow type could not be

formed. Therefore, the flow in this area is pulled upward by the flow near the wall from

falling down and driven by the shear. Therefore, with this distribution of F, it implies

the liquid phase velocity peak could be observed around the point xp where the sign

change of F occurs, and the M-shaped Wl profile. The following flow types were also

inferred for the upward turbulent bubbly flow in the large square duct as shown in Fig.

5.4. Due to the bubbles accumulation near the wall, the lighter mixture layer was

formed and pushed upward faster by the pressure gradient. In the core region, the liquid

is heavier and the flow is pulled upward by the faster layer near the wall. In other words,

two layers with the different flow types are formed between the wall to the duct center

including the upward pressure driven flow near the wall, and the viscous shear driven

flow in the core region. Wall retarding effect was confined in the lighter mixture layer or

blocked from transferring into bulk region. Moreover, if xp gets closer to the wall, the

wall effect is confined in a smaller area and blocked more.

To verify the above viewpoints, the corresponding lateral distributions of liquid

phase velocity were shown in Fig. 5.2(b) and Fig. 5.3(b) along the bisector line and

diagonal line for the bubbly flow, respectively. It is confirmed that once the local net

driving force F has an obvious sign changing near the wall, the lateral distribution of the

84

axial liquid-phase velocity Wl behaves as M-shaped. Besides, the location of the

velocity peak is observed to be lied close to xp according to the current experimental

data. Therefore it is suitable to choose F as an index of the flow type.

For the cases with the larger void fraction, since the bubbles coalescence occurs

and the bubbles were migrated to the core region, the net driving force F is negative

between xp and the wall, but positive between xp and the duct center. It indicates the net

viscous force acts as the driving force upward between the point xp and the wall, but the

flow resistance between xp and the duct center. Similarly, two layers with different flow

types as shown in Fig. 5.5 were also formed between the wall to the duct center but with

the viscous shear driven flow near the wall pulled by the faster flow in the duct center.

With further increasing the void fraction, more bubbles moved to the core region which

led to the heavier fluid near the wall. Once the viscous shear force is not strong enough

to overcome the local gravity, the W-shaped velocity profile with the downward flow

near the wall as shown in Fig. 5.1(d) is possible to be observed. However, so far the

current experimental technique (the X type hot-film anemometry used here) can’t

measure the liquid phase velocities under these flow conditions due to the limitations of

the hot-film anemometry measurement due to the high void fraction. Therefore,

currently it is lack of the experimental data of liquid velocities to validate the W-shaped

velocity profile.

In addition, for the intermediate void fraction cases in the transition regime such

as Jl=0.75m/s and Jg=0.18m/s and Jl = 1.0m/s, Jg = 0.27m/s, it is notable that along the

diagonal line, double peaks of F were observed as shown in Fig.5.3 (a) and 5.3(c) due to

the double void peak. For these cases, the peak locations of the liquid velocities tend to

migrate to the duct center especially along the diagonal line. Along the bisector line, the

peak locations of liquid velocities deviate more from xp due to the following reasons. As

85

we have confirmed, for the bubbly flow in this large square duct under certain condition,

peaks of void fraction and liquid phase velocity could be observed near the wall center

and corner with more apparent peak near the corner. For the small void fraction cases,

the locations of F=0 along the diagonal Xp1 and bisector Xp lines are close to each other.

However, for the cases in the transition regime such as Jl = 0.75m/s, Jg = 0.18m/s and at

Jl = 1.0m/s, Jg = 0.225m/s, Xp1 and Xp tends to be farther to each other, with Xp1 closer to

the duct center as shown in Fig. 5.6. And thus the location of the velocity peak along the

bisector line might be affected by that along the diagonal line and gets closer to the duct

center.

86

Fig. 5.2 Distributions of the local net driving force F for (a) Jl=0.75m/s and (c) Jl=1.0m/s and corresponding liquid velocities Wl for (b)

Jl=0.75m/s and (d) Jl=1.0m/s along the bisector line.

0 10 20 30 40 50 60 70-1500

-1000

-500

0

500

1000

1500

Distance from the wall x (mm)

Lo

cal

net

dri

vin

g f

orc

e F

0.09

0.135

0.18

0.27xp

(a)

Jl=0.75 m/s

Jg (m/s)

0 10 20 30 40 50 60 700.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Liq

uid

vel

oci

ty W

l(m/s

)

0.0

0.09

0.135

0.18xp

(b)

Jg (m/s)

Jl=0.75 m/s

0 10 20 30 40 50 60 70-1500

-1000

-500

0

500

1000

1500


Lo

cal

net

dri

vin

g f

orc

e F

0.09

0.135

0.18

0.225

0.315xp

(c)

Jl=1.0 m/s

Jg (m/s)

0 10 20 30 40 50 60 700.7

0.8

0.9

1

1.1

1.2

1.3

1.4


Liq

uid

vel

oci

ty W

l (m

/s)

0.0

0.09

0.135

0.18

0.225xp

(d)

Jl=1.0 m/s

Jg (m/s)

high void fraction

high void fraction

low void fraction

low void fraction

87

Fig. 5.3 Distributions of the local net driving force F for (a) Jl=0.75m/s and (c) Jl=1.0m/s and corresponding liquid velocities Wl for (b)

Jl=0.75m/s and (d) Jl=1.0m/s along the diagonal line.

0 10 20 30 40 50 60 70-2000

-1500

-1000

-500

0

500

1000

1500


Lo

cal

net

dri

vin

g f

orc

e F

0.09

0.135

0.18

0.27

(a)

xp1

xp2

Jg (m/s)

Jl=0.75 m/s

0 10 20 30 40 50 60 700.4

0.6

0.8

1

1.2

1.4


Liq

uid

vel

oci

ty W

l(m/s

)

0.0

0.09

0.135

0.18xp1

xp2

(b)

Jl=0.75 m/s

Jg (m/s)

0 10 20 30 40 50 60 70-1500

-1000

-500

0

500

1000

1500

2000


Lo

cal

net

dri

vin

g f

orc

e F

0.09

0.135

0.18

0.225

0.315

xp1 x

p2

(c)

Jl=1.0 m/s

Jg (m/s)

0 10 20 30 40 50 60 70

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6


Liq

uid

vel

oci

ty W

l(m/s

)

0.0

0.09

0.135

0.18

0.225xp1 x

p2

(d)

Jl=1.0m/s

Jg (m/s)

high void fraction

low void fraction

low void fraction

high void fraction

88

Fig. 5.4 Schematic diagram of flow image for the small void fraction: (a) Bubble lateral

distribution; (b) Inside the bubble layer near wall; (c) Outside the bubble layer.

Fig. 5.5 Schematic diagram of flow image for the large void fraction: (a) Bubble

lateral distribution; (b) Flow near the wall; (c) Flow in the core region.

Fig. 5.6 Schematic diagram of velocity peak location.

89

5.3 Conditions for M-shaped Wl profile

As mentioned previously, except in the large square duct, in the existed database

of upward turbulent bubbly flow, there were very few observations of such significant

liquid-phase velocity peak near the wall region except that slightly peak or slightly shift

of velocity peak from the pipe center which is termed as ―chimney effects‖[5.3]. In the

small pipe, generally the steeper ―∩‖ shape or slightly shift of velocity peak was

observed even with significant wall peak of void fraction. Following the analysis in

section 4.3.1, in this section the conditions required for the formation of M-shaped Wl

profile will be discussed.

Firstly, based on the fully-developed flow the time averaged force balance

globally, we have

HggdzdP wgmglml / , (5.2)

where αlm=1-αgm and αgm are the mean volume fraction of the liquid and gas phases,

respectively. ηw is the wall shear stress per unit area exerted on the flow and is negative

if the wall drags the flow, and is also important to determine the internal flow structures.

H is the geometry dimension depending on the different geometry, and given for several

geometry in Table 5.2.

Table 5.2 Geometry dimension H to determine the wall shear stress

Duct

Geometry

Circular pipe

DH: pipe diameter

Channel

DH: Channel width

Square duct

DH: Duct width

Triangle

DH: Side length

H DH /4 DH /2 DH /4 DH /12

Substituting Eq. (5.2) to Eq. (5.1), the local net driving force F could be

rearranged as follows:

HgF wggmGl / . (5.3)

90

By setting Eq. (5.3) equals to zero, the sign changing point of F satisfies

gHGlwgmg 0 , (5.4)

where αg0 is the void fraction at the position where F changes the sign. Since αg

0 > 0, it

also requires

gHGlwgm . (5.5)

According to the previous analysis in Section 4.3.1, the M-shaped Wl profile is

possible to be formed when

pxx ,0 , 0

gg and cp xxx , , 0

gg , (5.6)

where xp is the distance between the location of the sign changing point of F and the

wall x=0. Besides, it is easier to observe M-shaped velocity profile if xp is closer to be

zero, since the wall effect was confined in a narrower region near the wall.

Eqs. (5.5) and (5.6) indicate that under the similar wall shear stress ηw/H, the

cases with larger αgm and steeper profile of the void fraction is easier to form M-shaped

velocity profile. Under the similar void fraction distribution, the cases with the smaller

wall shear stress ηw/H are easier to form the M-shaped velocity profile. The final flow

type is determined by the lateral void fraction distribution and the wall drag resistance.

Under the same flow conditions, duct geometry plays a dominated effect on determining

these two factors.

5.4 Duct geometry effect

To answer the reason why the typical M-shaped velocity profile is observed in

the large square duct rather than the small circular and non-circular ducts and also the

large circular pipe, the duct geometry effect on the void fraction distribution and the

wall shear stress is discussed in this section by examining the ability of gathering

bubbles to near the wall region and the wall shear stress under the similar liquid and gas

91

flow rate Jl and Jg.

For the upward turbulent bubbly flow with the wall peak of the void fraction, the

ratio γ between the maximum void fraction near the wall αgmax and that in duct center αgc

is defined as the peaking factor of the bubble gathering.

gcg /max . (5.7)

The larger γ indicates stronger bubble gathering near the wall. Generally, γ is

determined by the flow rate, bubble size, and duct geometry and so on. To show the

geometry effect, the flow conditions with the similar flow rates and bubble size were

chosen for the different experiments as shown in Table 5.3. Fig. 5.7 shows the

comparison of γ for the different duct geometries including small and large circular

ducts and noncircular ducts. The following results are obtained: (1) for the upward

turbulent bubbly flow in the circular ducts, γ for the small pipes γsc is larger than that for

the large pipes γlc, indicating stronger bubbles gathering ability in the smaller pipes; (2)

γln in the large noncircular duct is larger than that the large pipes γlc and especially near

the corner, where γln gets close to that for the small circular pipes γsc, indicating stronger

bubble gathering ability as compared to that in the large circular pipe due to the corner.

The duct geometry effect on the wall shear stress for two-phase flows is

examined by the well-known Lockhart-Martinelli correlation [5.9]. It shows the duct

geometry effect on the wall shear stress for the upward turbulent bubbly flow could be

estimated based on the single-phase flow under the same flow rate and a multiplier as

follows:

lwLtw ,

2

, , (5.8)

where 2

L is the frictional multiplier for two-phase flow and ηw,l is the single-phase wall

shear stress resulting from the liquid phase flow rate of Jl/αl. Next the duct geometry

effect on 2

L and ηw,l will be considered respectively to show its effect on the wall shear

stress for two-phase flows.

For the upward turbulent bubbly flow in the smooth pipe, according to

92

Chisholm-Larid correlation [5.9],

22 1211 XXL (5.9)

and gwlwX ,, / , (5.10)

where ηw,g is the single-phase wall shear stress resulting from the gas phase flow rate of

Jg/αg.

For the single-phase flow in the pipe, the wall shear stress ηw,l is usually

predicted based on the following empirical correlation:

2

, / 2w l l l lf W , (5.11)

where Wl is the mean flow rate of the single-phase flow; fl is the friction factor, and

could be estimated by fl=16/Rel and fl=0.0791/Rel0.25

for laminar and turbulent flow

respectively; Rel is Reynolds number, Rel=ρlVlDH/μl.

For the upward turbulent bubbly flow, substituting Eq. (5.11) into Eqs. (5.9) and

(5.10) for the liquid and gas phases to calculate the multiplier, it shows the duct

geometry has no effect on 2

L . Therefore, the duct geometry effect on the wall shear

stress for the upward turbulent bubbly flow could be estimated based on the

single-phase flow under the same flow rate where the larger duct corresponds to the

smaller wall shear stress. It means that the large square duct owns not only the stronger

bubble gathering ability to the wall and corner as compared with other large circular

ducts but also less wall shear stress compared with other smaller ducts. According to the

analysis in Section 5.2, it showed the flow with smaller wall shear stress, and steeper

void fraction profile is easier to form M-shaped velocity profile. Therefore, the

observation of the significant M-shaped Wl profile in the large square duct could be well

understood.

93

Table 5.3 Experimental conditions to for different geometries in the literatures. In the

table, C: circular duct; T: triangle duct; S: square duct; B: base length; H: height.

Exps.

Authors

Duct

shape

DH

(mm) Jl (m/s)

DB

(mm)

z

(DH)

Serizawa et al.[5.2]

C 60 1.03 3.5~4 20

Liu et al.[5.5]

C 57.2 0.75

1.083 2~4 36

Wang et al.[5.4] C 57.15 0.71

0.94 × 35

Shawkat et al.[5.6]

C 200 0.68 3~6 42

Sun et al.[5.7] C 101.6 1.021 4~5 18

Lahey et al.[5.8] T 50(B)

100(H) 1.0 5 73

Sun[5.1]

S 136 0.75

1.0 3~5 16

94

Fig. 5.7 Comparison of γ for different duct geometries for Jl around (a) 0.75m/s and (b)

1.0m/s. For noncircular geometry, ―-1‖ and ―-2‖ indicate data obtained near the wall

center and corner, respectively.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.351

2

3

4

5

6

7

8

9

c

Serizawa et al

Wang et al

Liu et al

Sun et al

Lopez et al-1

Sun et al-1

Lopez et al-2

Sun et al-2

Small circular pipe

Noncircular pipe

Large circular pipe

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.351

2

3

4

5

6

7

c

Wang et al

Liu et al

Shawkat et al

Sun et al-1

Sun et al-2

Small circular pipe

Large circular pipe

Noncircular pipe

(a)

95

5.5 Peak location of Wl near the wall

This section will discuss the peak location xl of the liquid phase velocity Wl near

the wall. At the peak location xl, the viscous force ηv is zero. To get xl, the lateral

distribution of ηv could be calculated on the bisector line by integrating of –F from the

wall or the duct center to the specified location x, respectively.

2/

2/0

H

H

D

xD

x

wv xdFxdF . (5.12)

For the fully-developed flow, ηv,D/2 = 0 due to the flow symmetry in the duct

center. Substituting Eqs. (5.1) and (5.3) to Eq. (5.12), and after rearranging, the lateral

distribution of ηv could be obtained as

1

the bubble blocking effect

2

1 2 / / 1

/ 2 ,

v w H l lm lm lm

l lm H

x D g

P g x D

, (5.13)

where αlm1 is mean void fraction between location x and wall; αlm2 is mean void fraction

between location x and center. The second term on the right hand shows the blocking

effect of bubble layer near the wall on wall retarding effect from transferring into the

core region. With M-shape velocity profile, for the fully developed flow, zero local

viscous force ηv is reached at the location xl due to the zero velocity gradients. According

to Eq. (5.13), at the location xl, we have 2 /lm lP g indicating it lies between

the wall and that of sign changing point of F. However, as shown from Figs. 5.2 and 5.3,

the velocity peak was between the sign changing point of F and the duct center, which is

not consistent with this analysis.

5.6 Flow characteristics in two layers

With the M-shaped liquid phase velocity profile, two layers with the different

flow types (pressure driven and shear driven flows) could be formed. In this section, the

flow characteristics in these two layers are presented based on the normalization of

96

liquid phase velocity.

Fig. 5.8 Definition of two layers.

5.6.1 Definition of two layers

Fig. 5.8 shows the definition of two layers divided by the peak of Wl. In the

inner layer where the wall strongly affects the entire region, the following

dimensionless quantities are defined,

lmWWWin /* , and mxxX in /* . (5.14)

The maximum magnitude of the velocity Wlm and the distance of this point to the

wall xm were chosen as the characteristic velocity and length for the different flow

conditions, respectively. As for the single-phase pipe flow driven by the constant

pressure gradient, the quadratic relation W*in =1- (X

*in -1)

2 could be derived in between

W*in and X

*in for the fully-developed laminar flow regime.

In the outer layer, due to symmetry at the center of the wall where the velocity

gradient is zero, there might exist an inflection point in the liquid velocity profile

between xm and xc as shown in Fig. 5.8. In the present study this feature will be not

focused due to the lack of the experimental data. In the outer layer where the wall

effects were blocked by the bubble layer, the following dimensionless quantities are

97

defined,

lclmlc WWWWWout *, and mc xxxxX out m

*. (5.15)

Herein the flow is driven by the local viscous shear generated by the faster inner

layer. The velocity difference (Wlm-Wc) and the distance (xc-xm) between the velocity

peak location and the duct center were chosen as the characteristic velocity and the

length for the different flow conditions, respectively. For the single-phase pipe flow

completely driven by the constant local net viscous shear dηv/ds (ηv: local viscous force

and s: coordinate direction, x or y), the quadratic relation W*out =(X

*out -1)

2 could be

derived in between W*out and X

*out for the fully-developed laminar flow regime. If the

flow is completely driven by the velocity difference like the Couette flow, the linear

relation W*out =1-X

*out could be derived for the fully-developed flow regime.

5.6.2 Velocity characteristics

By using the above normalization quantities and the measured experimental data

of the bubbly flow in the square duct [5.1], the velocity characteristics in the inner and

outer layers are shown along both the bisector and the diagonal lines in Fig. 5.9.

Because of the limitation of the pipe length in the experiments, the experimental cases

with the better developed M-shaped Wl profile was chosen. Besides since in Sun’s

experiments only a few points were measured in the inner layer, the cases of the upward

turbulent bubbly flow in the transition region were chosen. Considering the features of

the single-phase flow, it is treated all in the inner layer. In order to have an intuitive

understanding, the distributions of void fraction along both the bisector and the diagonal

lines could be found in Fig. 2.10 in Chapter 2.

In the inner layer, the velocity profile deviates from the quadratic function due to

the turbulent effects as shown in Figs. 5.9(a) and 5.9(b). Along the bisector line, it is

noted that W* under Jl=1.25m/s and Jg=0.27m/s becomes lager than the other cases. It

might be attributed to the following two reasons. Due to the high liquid superficial

velocity of this case, (1) the location of the velocity peak is closer to the wall; (2) the

98

flow might be less developed based on the uniform inlet flow distribution. In addition,

along the diagonal line near the corner it is observed that the velocity profile returns to

the quadratic function with the increase of the void fraction as shown in Fig 5.9(b). Two

possible reasons could be considered for this phenomenon. One is that the bubble

clustering region in these cases could be considered as a boundary which confined the

liquid flow between the wall and itself. It results in the smaller characteristic length as

compared with the single-phase flow. Therefore, the inertial effects in this layer will be

suppressed and the flow tends to be relaminarized. The other is that with the increase of

the void fraction, the flow in the inner layer also gets closer to the Couette flow, which

is completely driven by the velocity difference.

In the outer layer, the velocity profile is observed between the quadratic function

and the linear function as shown in Figs. 5.9(c) and 5.9(d). If much more bubbles are

accumulated near the wall and/or the corner as compared to the core region (steeper

void fraction distribution from the wall to the duct center), the velocity profile gets

closer to the Couette flow as shown in Fig. 5.9(d). It could be imagined that the

asymptote of the Couette flow could be reached for completely stratified flow of the gas

and liquid with the gas flow near the wall and the liquid flow in the duct center. In the

real situation, with the increase of the void fraction, the coalescence of the bubbles

occurs which makes the bubble migrate to the duct center region. Under the large

enough void fraction, considering the mixture of the fluids, the similar tendency to the

Couette flow will be formed between the duct center and near the wall region. Therefore,

it further confirms that for the upward turbulent bubbly flow in this large square duct,

two layers with the different flow types were easy to be formed. According to these

characteristics, the two-layer model which models the turbulent bubbly flow seperatedly

in different regions according to the different flow types was suggested to improve the

modeling of the upward turbulent bubbly flow therein.

In addition, unlike the flows in the circular pipes with the axisymmetric flow

distribution, for the single-phase turbulent flow in square duct there exists the secondary

flow in cross section due to the non-axisymmetric lateral turbulent intensities [5.10].

99

For the turbulent bubbly flow in the square duct, the non-axisymmetric void distribution

may also induce the horizontal secondary flow due to non-axisymmetric buoyancy

effect. The existence of the secondary flow could have a strong effect on flow

characteristics such as distorting the local mean velocity distribution. As shown in Fig.

5.9(d), it was observed that W*

out in the middle region, which might be related to the

convective transport of the axial liquid momentum due to the existence of secondary

flow. Therefore, in order to improve the model of the turbulent bubbly flow in the

non-circular ducts, the secondary flow effect should be considered. However, currently

there are no experimental information about the secondary flow of turbulent bubbly

flow therein, so the experimental measurements about the secondary flow is suggested

to carry out in the future work.

100

Fig. 5.9 Velocity characteristics in the inner layer along (a) the bisector line and (b) diagonal line; and in the outer layer (c) along the

bisector line and (d) the diagonal line. The mean void fraction could be approximated as αgm=Jg/(Jg+Jl).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(a)

Dimensionless Distance, X*

in

Dim

ensi

on

less

Liq

uid

velo

cit

y, W

* in

Symbol Jl, J

g

0.50 0.135

0.75 0.180

1.25 0.270

0.50 0.000

0.75 0.000

1.00 0.000

1.25 0.000

1-(X*-1)

2

turbulenteffect

Wall flowLaminar regime

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(b)


in

Dim

ensi

on

less

Liq

uid

velo

cit

y, W

* in

Symbol Jl, J

g

0.50 0.135

0.75 0.180

1.25 0.270

0.50 0.000

0.75 0.000

1.00 0.000

1.20 0.000

1-(X*-1)

2

Wall flowLaminar regime

Increase of

turbulenteffect

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(c)


out

Dim

ensi

on

less

Liq

uid

velo

cit

y, W

* out

Symbol J

l, J

g

1-X*

out

2

1-X*

out

0.75 0.090

0.75 0.135

0.75 0.180

1.00 0.090

1.00 0.135

1.00 0.180

1-Xout

*2

1-Xout

*

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(d)


out

Dim

ensi

on

less

Liq

uid

velo

cit

y, W

* out

Symbol J

l, J

g

1-X*

out

2

1-X*

out

Steeper

distribution

0.75 0.090

0.75 0.135

0.75 0.180

1.00 0.090

1.00 0.135

1.00 0.180

1-Xout

*2

1-Xout

*

Couette type flow

Couette

type flow

101

5.7 Numerical validation of the liquid phase velocity

In this section, the numerical validation of the liquid phase velocity profile was

carried out based on the experimentally measured pressure drop and the void fraction

distribution [5.1].

5.7.1 Numerical models

In order to obtain the lateral distribution of the axial liquid phase velocity W[,

the force balance equation in the axial direction is solved according to the formulation

proposed by Ishii and Hibiki [5.11]. For the gas phase:

, ,0.

g xz g g yz g

g g g DZ

dpg F

dz x y

(5.16)

For the liquid phase:

, ,1 1

1 0,g xz l g yz l

g DZ

dpF

dz x y

(5.17)

where dp/dz is the pressure gradient for the upward turbulent bubbly flow; ηxz,l, and ηxz,g

are the shear stress induced by the molecular and eddy viscosity for liquid phase and gas

phase, respectively; FDZ is the drag force due to the phase interaction; ρl and ρg are the

liquid and gas density, respectively; g is gravity acceleration: -9.81m2/s. By summing

Eqs. (5.16) and Eq. (5.17) and neglecting the gas phase due to ρg <<ρl,

, ,1 11 0.

g xz l g yz l

g l

dpg

dz x y

. (5.18)

Considering the geometrical symmetry, the governing equation on the bisector lines is:

11 0

L

g sz

g l

ddpg

dz ds

, (5.19)

where ds=dx or ds=dy for the bisector line vertical to y or x axis, respectively.

The one-equation turbulence model is used to express the liquid phase turbulent

102

Reynolds shear stress as ηsz,l=ρlvt dWl/ds in terms of the eddy viscosity vt and the liquid

phase velocity gradient dWl/ds. The turbulence eddy viscosity vt is expressed in the form

of t tv kl . k, lt and k are the turbulent kinetic energy, the characteristic length scale

and the velocity of turbulent structures for the upward turbulent bubbly flow,

respectively. Substituting the Reynolds stress ηsz,l to Eq. (5.19), it is obtained that

1

1 0

lg l t

g l

dWd kl

dp dsg

dz ds

. (5.20)

Eq. (5.20) was solved to get the liquid phase velocity profile Wl given by the

experimental measured pressure gradient, the void fraction distribution and the turbulent

kinetic energy. The flow continuity is maintained by setting the mean liquid flow rate.

For the upward turbulent bubbly flow, the turbulence could be induced by both

the liquid shear and the bubbles relative motion. When the bubble-induced turbulence

played a dominated role, the characteristic length scale lt could depend on the bubble

diameter DB and also the bubble-bubble interval distance lB-B due to the interactions

between the bubble-induced eddies. Here, the mean vertical bubble-bubble interval

distance lB-B is chosen as the flow characteristic length scale lt during solving Eq. (5.20).

As shown in Fig. 5.10, lB-B could be estimated based on the mean bubble diameter DB

and the void fraction αg as follows:

ggBN

BLBBB DNTWlB

111

. (5.21)

Fig. 5.10 Estimation of lB-B from the experiments. T, TB and TL are the total measuring

time, bubbles passing time and liquid passing time, respetively. WB is the mean bubble

velocity.

103

5.7.2 Numerical results

Fig. 5.11 shows the comparison between the predicted axial liquid phase

velocity Wl and the experimental measurement on the bisector line. It is observed that:

(1) with the characteristic length scale of lB-B, the liquid phase velocity profile Wl could

be predicted very close to the experimental measurement especially in core region,

indicating lB-B could be another suitable parameter to model the bubble-induced

turbulence for the upward turbulent bubbly flow. However, it still needs further

validations based on more experimental database. (2) As compared to the experimental

data, the predicted velocity profile Wl shows the flow all in the outer layer where the

flow was driven by shear. The lateral distribution of the mean viscous force ηv calculated

from the experiments confirms this point as shown in Fig. 5.12, since it is all negative.

The two assumptions are attributed for the disagreement with the experimental

measurements, that is, neglecting the flow acceleration and the uniform pressure

gradient distribution among the cross sections. However, in the experiments though the

void fraction distribution firstly reached the developed status at z = 4.5DH, the

liquid-phase velocity distribution was not confirmed whether it reached the

fully-developed status or still on the developing way at z = 16DH [5.1] since no lateral

downstream measurements were carried out. If the velocity was still on the developing

way, local flow acceleration term should be considered under the uniform pressure

gradient distribution. Therefore, to improve the upward turbulent bubbly flow modeling

in the noncircular ducts, it is necessary to establish the experimental database with

better developed-flow status to validate current models. Moreover the experiments with

the detailed flow developing process are also necessary to model the flow acceleration.

104

Fig. 5.11 Comparison between numerical predicted liquid phase velocity Wl and

experimental measurement on the bisector line for (a) Jl =0.75m/s and (b) Jl =1.0m/s,

respectively.

Fig. 5.12 Lateral distribution of mean viscous force ηv based on the experimental

measurements for (a) Jl =0.75m/s and (b) Jl =1.0m/s, respectively.

0 10 20 30 40 50 60 700.75

0.8

0.85

0.9

0.95

1

1.05

Distance from the wall x(mm)

Wl

Prediction (Jg=0.09)

Experimental data


Experimental data

Jl= 0.75m/s

(a)

0 10 20 30 40 50 60 701.05

1.1

1.15

1.2

1.25

1.3

1.35

Distance from the wall x(mm)W

l


Experimental data


Experimental data

(b)

Jl= 1.0m/s

0 10 20 30 40 50 60 70-15

-10

-5

0


Vis

cou

s sh

ear

stre

ss

v

(a)

Jl = 0.75m/s

Jg (m/s)

0.09

0.135

0 10 20 30 40 50 60 70-20

-15

-10

-5

0


Vis

cou

s sh

ear

stre

ss

v

(b)

Jl = 1.0m/s

Jg (m/s)

Jg0.09

Jg0.135

105

5.8 Summary and discussion

In this section, the formation of the typical M-shaped velocity profile of the

upward turbulent bubbly flow in the large square duct was studied by considering the

index of flow type, the formation conditions and the peak locations of the velocity

profile. The following findings were obtained therein.

(1) Neglecting the fluid acceleration and secondary flow, the local net driving

force F could be chosen as an index to show the flow type. Based on the sign

of F the formation of the M-shaped or W-shaped velocity profile could be

implied for the turbulent bubbly flow with the small and large void fractions

respectively, which has been verified for the small void fraction cases.

(2) According to the sign of F, it was found that for the turbulent bubbly flow in

the large square duct two layers with the different flow types could be

formed between the wall and the duct center. For the small void fraction

cases, two layers are the upward pressure driven flow near the wall, and the

viscous shear driven flow in the core region. For the large void fraction cases,

two layers are the upward pressure driven flow in the core region, and the

viscous shear driven flow near the wall.

(3) According to the analysis on the conditions for the formation of the

M-shaped velocity profile, the final flow type was determined by the lateral

void fraction distribution and the wall drag resistance, where the duct

geometry plays a dominated role on determining these two factors.

(4) The phenomenon of the M-shaped velocity profile in the large square duct

could be attributed to the geometry effect (a large square duct) which owns

not only stronger bubble gathering ability to the wall but also the smaller

wall shear stress.

(5) As for the peak location of the velocity, based on the current knowledge it

lies between the wall and that of sign changing point of F. However, the

experimental measurements of the velocity profile showed an inconsistent

106

result.

(6) According to the flow characteristics, the two-layer model for the upward

turbulent bubbly flow in the large square duct was suggested. The current

numerical validation indicates that the bubble-bubble distance could be

another important parameter to model the bubble-induced turbulence.

It is notable that the above analyses were carried out assuming the flow reached

the developed status without the secondary flow and uniform pressure gradient in the

cross section. Even though the experimental data at z = 16DH was used with better

developed status according to the analysis in Section 4.2, there are no further

measurements after z = 16DH to verify the flow reached the developed status or not. As

is known, the secondary flow among the cross section is the typical characteristics for

the single-phase turbulent in the noncircular ducts as compared to that in the circular

pipes. Here this effect was assumed to be neglected because of the following reason.

The experimental data of the lateral turbulent normal stresses (<uu> and <vv>) in [5.2]

showed that more symmetric as compared with that of the single-phase flow due to the

formation of the bubble layer near the wall and corner for the turbulent bubbly flow in

the square duct. It indicated the horizontal secondary flow may be very weak and

mainly exist in the region very near the corner; and the secondary flow could be

neglected. However, since there is no direct experimental information about the

horizontal convection and there may exist some error of the lateral normal turbulent

stresses (<uu> and <vv>), it still needs further consideration about this assumption.

Besides, the analysis implied the W-shaped velocity profile for the large void

fraction cases, the lack of the experimental data limited us the further validation. In

addition, the inconsistence about the peak location might be attributed to the

assumptions used. Therefore, in order to understand and develop the model of the

turbulent bubbly flow in the large square duct, the further detailed measurements should

be carried out with checking the flow developing status. The measurements of the

secondary flow were also suggested for the modeling of the turbulent bubbly flow in the

noncircular duct.

107

Reference







[5.4] Wang, S., Lee, S., Jones, O., Lahey, R., 1987, 3-D turbulence structure and


Flow 13, 327–343.



Transfer 36, 1049–1060.






[5.8] Lahey, Jr R. T., Bertodano, de M. L., Jones, Jr O. C., 1993, Phase distribution in

complex geometry conduits, Nucl. Eng, Des. 141, 177-201.

[5.9] Lockhart, R. W., Martinelli, R. C., 1949, ―Proposed Correlation of Data for

Isothermal Two Phase Flow, Two Component Flow in Pipes,‖ Chem. Eng. Prog.,

45, pp. 39–48.

[5.10] Hoagland, L. L., 1960, Fully Developed Turbulent Flow in Straight Rectangular

Ducts - Secondary Flow, Its Cause and Effect on the Primary Flow, Ph.D. thesis,

Department of Mechanical Engineering, MIT.

[5.11] Ishii, M., Hibiki, T., 2012, Thermo-Fluid Dynamics of Two-Phase Flow,

Springer.

108

109

Chapter 6

Void fraction of the turbulent bubbly flow

6.1 Introduction

As mentioned previously, for the turbulent bubbly flow, the deformation and

size of the bubble could significantly affect the bubbles’ behavior as well as the

associated momentum, heat and mass transfer processes. In the numerical predictions of

the turbulent bubbly flow, they are essential to calculate the bubble drag and lift forces,

which are two important forces to determine the fluid velocities, the phase distributions

and so on [6.1-6.4]. So far, a number of studies have been carried out on the effects of

the bubble’s deformation and size on the drag and lift force coefficients, CD and CL,

most of which were based on the single bubble experiments in different flow systems,

especially in the stagnant and simple shear flow systems [6.5-6.10]. For the bubbly flow,

the bubble size has long been an important measurement to understand the bubbly flow

characteristics with the aid of the optical or electrical probe techniques [6.11-6.13].

Using the general optical and electrical probe techniques, the time-averaged chord

length Lc of the bubbles could be measured, and indicate the characteristic axial size of

the bubbles. Based on the spherical bubble assumption, the volume equivalent spherical

bubble diameter (Sauter mean diameter: DSM ) was estimated by DSM =1.5Lc.

As for the measurement of the bubble’s deformation, the flow visualization is an

efficient way, and some correlations have been established based on the experiments of

the single bubble or droplet in the infinite stagnant liquids or various conduits [6.5].

However, since the flow visualization fails for the bubbly flows due to the low visibility,

there are few researches on the bubbles’ deformation for the bubbly flow except under

the low void fraction [6.14]. Currently, the numerical models of the bubbly flow mainly

110

consider the bubble deformation based on the single-bubble correlations [6.2, 6.8].

On the other hand, based on the double-sensor or multi-sensor probe, the

interfacial area concentration (IAC) ai , the void fraction αg, the averaged chord length

of the bubbles Lc could be measured for the bubbly flow simultaneously. Coupling the

IAC with the local time-averaged void fraction αg, the Sauter mean diameter DSM could

be given as follows:

DSM =6αg/ ai. (6.1)

With these three variables, it is regarded here that the bubble deformation for the

bubbly flow could be statistically reflected by combing the Sauter mean diameter DSM

and chord length Lc. In this chapter, the bubble deformation will be firstly studied in the

turbulent bubbly flow including many bubbles based on the recently established

experimental database by Sun [6.15]. After that, the numerical validation of the void

fraction will be carried out based on the experimental measurements.

6.2 Bubble deformation

6.2.1 Definition of bubbles’ statistical shape

To begin with, the time-averaged Sauter mean diameter DSM and chord length Lc

measured from the experiments were firstly shown in Figs. 6.1-6.3 as a database along

the bisector, diagonal and the near wall lines, respectively. The current analysis was

limited for the bubbles with the time-averaged DSM ranging from 3.5mm~5.5mm and Lc

ranging from 2.5mm~4.0mm. The following trends could be observed:

(1) When it gets close to the wall, Lc shows monotonously increase, but DSM

shows different tendency, indicating the effect of the bubble deformation changing from

the duct center to the near wall region. Since the present study mainly considers the

void fraction effects on the bubble deformation, the following comparisons will be

mainly carried out in the core region and neglecting the wall effects.

(2) DSM doesn’t change with increase of the gas flow rate or the void fraction,

111

but Lc clearly increases. According to the definition of Lc, the characteristic length of the

bubbles in the axial direction could be represented by Lc even for the different bubble

shapes [6.16]. Therefore, for the bubbly flow the change of Lc under the same DSM

implies the change of the bubble deformation statistically, which will be analyzed in

this section.

To investigate the bubble deformation for the bubbly flow, the shape factor of

the bubbles is defined and examined in this part. From the flow visualization for

different flow conditions, the bubbles could be assumed to be the ellipsoidal shape with

circular plane vertical to the axial direction from the statistical point of view, as shown

in Fig. 6.4. DV and DH are defined as the dimensions in the axial and horizontal

directions, respectively. Liu and Clark [6.16-6.17] have carried out a series of studies on

the relation between the chord length and the axial dimension of the bubbles for the

different shapes. Assuming a spherical or ellipsoidal bubble, DV could be calculated as

DV=1.5Lc if there were enough bubbles penetrated by the sensor.

So far, by a four-sensor optical probe, the void fraction, the IAC and the chord

length were measured. Assuming to the bubbles to be spherical, DSM could be calculated.

Similarly, assuming the bubbles to be statistically ellipsoids, it is possible to calculate

the bubble deformation based on DV=1.5Lc as follows.

Firstly, the shape of the bubbles is defined as the aspect ratio E between the

bubble horizontal and axial dimensions,

VH DDE / , (6.2)

Based on E, the volume VB and the surface area SB could be calculated by

66

322

VVHB

DEDDV

, (6.3)

and 23

732

22

12 EE

VB

EDS

. (6.4)

Then, the bubble deformation could be obtained by

112

23

7321

26

EEBV

B

SD

V

. (6.5)

According to the definition of DSM (iB

B

aS

V 66 ) and substituting DV = 1.5Lc, Eq.

(6.5) can be rearranged as

23

7321

2

5.1EEc

SM

L

Dr

. (6.6)

With DSM and Lc, E could be shown by the ratio between DSM and LC measured by the

four-sensor optical probe. After solving Eq. (6.6), it could be obtained as

2

422

624

843402

r

rrrE

. (6.7)

The relation between E and DSM/LC was plotted in Fig. 6.5. If DSM/LC is larger

than 1.5, it indicates E>1 which corresponds to the statistical oblate bubbles. If DSM/LC

equals to 1.5, it indicates E=1 which corresponds to the statistical spherical bubbles.

Otherwise, it indicates the statistical prolate bubbles. Therefore, the statistical bubble

shape could be summarized based on DSM/LC as

prolateFor 1.5

sphericalFor 5.1

oblateFor 5.1

c

SM

L

D (6.8)

In the following section, DSM/Lc will be examined to show the bubble statistical

deformation.

113

Fig. 6.1 Distributions of the Sauter mean diameter DSM and the chord length Lc along the bisector line under different flow conditions :(a)

Jl =0.5m/s; (b) Jl =0.75m/s; (c) Jl =1.0m/s; (d) Jl =1.25m/s. The solid and dashed lines show DSM, and Lc respectively.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(a) Jl =0.5 m/s

Jg0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(b) Jl =0.75 m/s

Jg

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(c)J

l =1.0 m/s

Jg

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(d)J

l =1.25 m/s

Jg

0.135 0.18 0.225 0.27

114

Fig. 6.2 Distributions of the Sauter mean diameter DSM and the chord length Lc along the diagonal line under different flow

conditions :(a) Jl =0.5m/s; (b) Jl =0.75m/s; (c) Jl =1.0m/s; (d) Jl =1.25m/s. The solid and dashed lines show DSM, and Lc respectively.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(a) Jl =0.5 m/s

Jg

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(b) Jl =0.75 m/s

Jg

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(c) Jl =1.0 m/s

Jg

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(d) Jl =1.25 m/s

Jg

0.135 0.18 0.225 0.27

115

Fig. 6.3 Distributions of the Sauter mean diameter DSM and the chord length Lc along the near-wall line under different flow

conditions :(a) Jl =0.5m/s; (b) Jl =0.75m/s; (c) Jl =1.0m/s; (d) Jl =1.25m/s. The solid and dashed lines show DSM, and Lc respectively.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(a) J

l =0.5 m/s

Jg

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(b) Jl =0.75 m/s

Jg

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(c)J

l =1.0 m/s

Jg

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.072

2.5

3

3.5

4

4.5

5

5.5x 10

-3

x (mm)

DS

M&

LC

(d) Jl =1.25 m/s

Jg

0.135 0.18 0.225 0.27

116

Fig. 6.4 Schematic diagram of the ellipsoidal bubble shape.

Fig. 6.5 Aspect ratio E vs. the ratio DSM/Lc for the ellipsoidal bubbles.

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

DSM

/Lc

E

Oblate bubble

Prolate bubble

Spherical bubble

117

6.2.2 Distribution of bubbles’ shape factor

Together with the void fraction distributions, Figs. 6.6-6.8 show DSM/Lc for

different flow conditions along the bisector, the diagonal and the near-wall lines,

respectively. It is observed as follows:

(1) DSM/Lc decreases when it gets close to the wall, and for the low void

fraction cases, DSM/Lc is greater than 1.5 in the core region and sometimes

less than 1.5 near the wall where there are high local void fraction and

liquid shear. According to the definition of DSM/Lc in Eq. (6.8), it means the

bubbles statistically behave oblate in the core region, which is qualitatively

in agreement with that observed in the single bubble or droplet experiments

in the infinite liquids [6.5].

(2) However, increasing the void fraction, DSM/Lc decreases in the whole area

including both the core region and near the wall, indicating that the bubbles

statistically behave more elongated. Especially for the largest void fraction

case here, the bubbles show statistically prolate even in the core region, and

the most prolate bubbles could be implied near the corner, which coincides

with the prolate bubbles cluster observed by the flow visualization.

The above analysis indicates that for the bubbly flow with the different flow

conditions the bubbles don’t always behave oblate even in the core region. In addition,

the existence of the many bubbles may affect the bubble deformation to deviate from

the single bubble case. However, before considering the void fraction effects, it is

necessary to examine whether the above observation is owing to the variation of the

bubble size for the different flow conditions or not. In the following, whether and how

the many bubbles affect the bubble deformation will be focused on.

118

Fig. 6.6 Distributions of the ratio r along the bisector line under different flow conditions: (a) Jl =0.5m/s; (b) Jl =0.75 m/s;

(c) Jl =1.0 m/s; (d) Jl =1.25m/s.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(a) Jl =0.5 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(b) Jl =0.75 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(c) Jl =1.0 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(d) Jl =1.25 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.135 0.18 0.225 0.27

119

Fig. 6.7 Distributions of the ratio r along the diagonal line under different flow conditions: (a) Jl =0.5m/s; (b) Jl =0.75m/s;

(c) Jl =1.0m/s; (d) Jl =1.25m/s.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(a) Jl =0.5 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(b) Jl =0.75 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(c) Jl =1.0 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(d) Jl =1.25 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.135 0.18 0.225 0.27

120

Fig. 6.8 Distributions of the ratio r along the near-wall line under different flow conditions: (a) Jl =0.5m/s; (b) Jl =0.75m/s;

(c) Jl =1.0m/s; (d) Jl =1.25m/s.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(a) Jl =0.5 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(b) Jl =0.75 m/s

Jg

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(c) Jl =1.0 m/s

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

1

1.5

2

DS

M/L

C

(d) Jl =1.25 m/s

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

x (mm)

Jg

0.135 0.18 0.225 0.27

121

6.2.3 Void fraction effects on shape factor

6.2.3.1 Bubble deformation for single-bubble case

As for the bubble deformation, theoretically it is governed by the force balance

on the bubbles from the continuous phase including the normal force, viscous shear

stress, gravity and the surface tension. The previous researches of the bubble

deformation in the single bubble or droplet experiments mainly considered the bubble

deformation correlated based on the dimensionless numbers including Eo, M, Re and

We, when it is dominated by different effects such as the surface tension, inertial or

viscosity effects.

For the bubble deformation of the ellipsoidal bubbles, Clift et al. [6.5]

recommended the correlation proposed by Wellek et al. [6.18] which analyzed

photographs of rising drops of immiscible liquids in continuous liquid. Therein, for low

M system with Eo < 40, the effect of Re is not important and the bubble deformation E0

for the single bubble could be estimated by

0.757

0 1 0.163H

V

DE Eo

D , (6.9)

or1/3

0H SMD D E . (6.10)

This correlation has been well validated based on the single-bubble experiments

[6.5, 6.19]. Currently, it is usually adopted in the two-fluid or multi-fluid model to

calculate the lift force coefficient CL which is related with the bubbles deformation and

is also crucial to the phase prediction of the numerical models [6.19]. Here, the bubble

size effect on the bubble deformation will be examined based on the above single

bubble correlation. By substituting Eq. (6.9) to Eq. (6.2) to eliminate the bubble size

effect, it is obtained that

122

0

1, other effects make the bubble more prolate;

/ 1, the same as the single bubble case;

1, other effects make the bubble more oblate.

R E E

(6.11)

Here, Eα is the statistical aspect ratio of the bubbles for the bubbly flow case.

Figs. 6.9-6.11 show the distribution of Eα/E0 along the bisector, the diagonal and the

near-wall lines, respectively. Based on the previous assumption and after eliminating

bubble size effect, it is observed that for the bubbly flow, the bubble deformation Eα was

over-predicted based on the single bubble correlation. Besides, with the increase of void

fraction, it deviates more from the single bubble correlation. Therefore, for the bubbly

flow, the correlation of the bubble deformation may need further correction considering

the void fraction effects.

123

Fig. 6.9 Distributions of the ratio Eα/E0 along the bisector line under different flow conditions: (a) Jl =0.5m/s; (b) Jl =0.75m/s;

(c) Jl =1.0m/s; (d) Jl =1.25m/s.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(a)J

l =0.5 m/s

Jg

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(b) J

l =0.75 m/s

Jg

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(c)J

l =1.0 m/s

Jg0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(d)J

l =1.25 m/s

Jg

0.135 0.18 0.225 0.27

124

Fig. 6.10 Distributions of the ratio Eα/E0 along the diagonal line under different flow conditions: (a) Jl =0.5m/s; (b) Jl =0.75m/s;

(c) Jl =1.0m/s; (d) Jl =1.25m/s.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(a)

Jl =0.5 m/s

Jg

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(b) Jl =0.75 m/s

Jg

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(c)

Jl =1.0 m/s

Jg

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(d) Jl =1.25 m/s

Jg

0.135 0.18 0.225 0.27

125

Fig. 6.11 Distributions of the ratio Eα/E0 along the near-wall line under different flow conditions: (a) Jl =0.5m/s; (b) Jl =0.75m/s;

(c) Jl =1.0m/s; (d) Jl =1.25m/s.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(a) Jl =0.5 m/s

Jg

0.045 0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(b) J

l =0.75 m/s

Jg

0.068 0.09 0.135

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(c) Jl =1.0 m/s

Jg

0.09 0.135 0.18 0.225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.5

0.6

0.7

0.8

0.9

1

x (mm)

E/E

0

(d) Jl =1.25 m/s

Jg

0.135 0.18 0.225 0.27

126

6.2.3.2 Correction on the multi-bubble effect

As compared to the single bubble case with the similar bubble size, the

introduction of the many bubbles may change the force balance exerted on the bubbles

which dominates the bubble deformation. One effect of the many bubbles is to

statistically decrease the relative velocity VR due to the increase of the mixture viscosity

μm or drag on the bubbles [6.20-6.21]. Assuming the simple similarity between the

single bubble system and bubbly flow system, the decrease of VR and increase of μm

could modify the bubble deformation by decreasing the inertial effects and increasing

viscous effects respectively. The above two effects could be reflected by the Reynolds

number. In the present study, the void fraction effect on the statistical bubble

deformation was considered according to Re in the following form,

0 0

Re

Re

R

S RS

VE E f E f

V

, (6.12)

where Reα = ρlVRαDSM/μl and ReS=ρlVRSDSM/μl are Reynolds numbers for the bubbly

flow case with the void fraction α and for the single bubble case, respectively; and VRα

and VRS are the relative velocities between the bubbles and liquid phase for the bubbly

flow case and single bubble case, respectively.

Based on the assumption of the similarity between the single bubble system and

bubbly flow system, Ishii and Zuber [6.20] proposed the following relative velocity VRα,

75.11 RSR VV , (6.13)

According to the measurements of the buoyancy-driven bubbly flow, Garnier et

al. [6.21] proposed the following relation between VRα and α,

3/11 RSR VV . (6.14)

In the present study, Eq. (6.13) is chosen. Substituting Eq. (6.13) to Eq. (6.12)

and after rearranging, it is obtained that

-1fEE S . (6.15)

127

To identify the function f(1-α), Fig. 6.12 shows the relation between Eα/ES and

(1-α) by collecting the experimental data under the different flow conditions. A clear

tendency of the void fraction effect on the bubble deformation was shown and the

function f(1-α) is obtained as follows,

3.2-1-1 f . (6.16)

For the single bubble case α=0, the above Eq. (6.15) becomes Eq. (6.9) which keeps the

consistency.

Replacing ES in Eq. (6.9) by Eα in Eq. (6.15), therefore for the bubbly flow, the

statistical horizontal dimension of the bubbles in Eq. (6.9) could be calculated by,

1/3 0.7670.7571 0.163Eo 1-H SMD D . (6.17)

6.2.4 Bubble size validation

Figs. 6.13 and 6.14 show the comparisons among the statistical horizontal

dimensions of the bubbles calculated from the experimental data DH.exp, DHS calculated

based on Eq. (6.9) and DHα calculated based on Eq. (6.17). It indicates that with ES, the

bubble deformations for the bubbly flow are all over-predicted, but with Eα the bubble

deformation for the bubbly flow could be predicted much better. Therefore, Eq. (6.17) is

recommended to calculate the statistical horizontal dimension DH for the bubbly flow.

However, it is notable that the above correlation Eqs. (6.15)-(6.17) was validated only

for the limited range of the bubble size around 3.5mm~5.5mm and the void fraction

around 5%~30%. For the bubbly flow with the different range of the bubble size and

void fraction, it still needs further validation and consideration based on more

experimental data in the future.

128

Fig. 6.12 Ratio R=Eα/E0 vs. the liquid fraction (1-α) under all the tested flow conditions.

Fig. 6.13 DH.exp vs. DHS under all the tested flow conditions.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.4

0.5

0.6

0.7

0.8

0.9

1

liquid fraction (1-)

R

experimental data

(1-)1.6

(1-)2.3

(1-)3

3 3.5 4 4.5 5 5.5 6

x 10-3

3

3.5

4

4.5

5

5.5

6x 10

-3

DHS

DH

,ex

p

experimental data

DH,exp

=DHS

DH,exp

=0.8DHS

129

Fig. 6.14 DH.exp vs. DHα under all the tested flow conditions.

6.3 Numerical validation of the void fraction

In this section, the numerical validation of the lateral distribution of the void

fraction will be carried out based on two-fluid model and the experimentally measured

turbulent intensities and the liquid-phase velocity distributions [6.15].

6.3.1 Numerical models

As shown in Chapter 5, in order to validate the liquid-phase velocity distribution

the force balance in the axial direction was solved based on the mixing length theory to

model the turbulent Reynolds shear stress. In this chapter, the force balance equation in

the horizontal direction will be solved in order to obtain the lateral distribution of the

void fraction following the method used in the small circular pipe [6.22]. Firstly,

assuming the fully developed flow status and neglecting the secondary flow effect, the

force balance in the horizontal direction is shown for the liquid and gas phases,

respectively. For the gas phase:

3 3.5 4 4.5 5 5.5 6

x 10-3

3

3.5

4

4.5

5

5.5

6x 10

-3

DH

DH

,ex

p

experimental data

DH,exp

=1.05DH

DH,exp

=DH

DH,exp

=0.95DH

130

0

G G

xx xy

Lx

P

X X Y

τ τM , (6.18)

0

G G

xy yy

Ly

P

Y X Y

τ τM . (6.19)

For the liquid phase:

1 1

1 0

L L

xx xy

Lx

P

X X Y

τ τM , (6.20)

1 1

1 0

L L

xy yy

Ly

P

Y X Y

τ τM , (6.21)

where MLx and MLy are the lift forces in X and Y directions, respectively. Eliminating

the pressure gradient by 20.6118.6 and 21.6119.6

and neglecting the turbulent stress of the gas phase due to 1LG , we have

1 10

L L

xx xy Lx

X Y

τ τ M, (6.22)

1 1

0

L L

xy yy Ly

X Y

τ τ M. (6.23)

With the above assumptions, the local void fraction distribution is determined by

the balance between the lateral turbulence dispersion force and the lift force. The lateral

turbulence dispersion is closely related to turbulence anisotropy which will be

considered in Chapter 7. Currently, the experimental measurements of the lateral

turbulence will be adopted.

Considering the geometrical symmetry, on the bisector and diagonal lines, Eqs.

(6.22) and (6.23) could be simplified as follows:

1

0

L

xx Lxd

dX

τ M =>

XGXF

dX

d

1

1 (6.24)

131

where duu

F XuudX

and L R LC W dWG X

uudX .

By solving Eq. (6.24) the lateral void fraction distribution could be obtained as

''

Gc0 0 0

Gc

1- 1- '' exp ' ' '' exp ' '

1- 1-

X y X

L R LcL LC

Lc

X G y F y dy dy F y dy

C W uuW W

uu uu

(6.25)

where αgc, LCW and Lcuu are the void fraction, liquid phase velocity, the normal

turbulent intensities at the duct center.

To solve Eq. (6.25), the lateral distributions of the liquid phase velocity and the

normal turbulent intensities are required in addition to αgc according to the experimental

databases. It is notable that in [6.15], the experimental measurements were carried out

based on the coordinate parallel with the wall direction. Here, during solving Eq. (6.25)

along the diagonal line, the relation about the turbulent normal components shown Fig.

6.15 is adopted.

The relative velocity WR is determined by the following correlation

4

3

l g

R

l

dW

g

. (6.26)

As for the lift coefficient CL, Tomiyama’s correlation [6.8] was used and therein

the bubble horizontal dimension DH was considered by 3/1757.0

0 163.01 EoDD SMH

and 767.03/1757.0 1163.01 EoDD SMH , respectively.

6.3.2 Numerical results

Firstly, Figs. 6.16-6.18 show the comparison of the lateral void fraction

distribution predicted by Eq. (6.25) and measured experiments at z=16DH along the

bisector and diagonal lines respectively. From these figures, it could be observed that:

(1) The predictions of the void fractions are in well consistent with the

132

experimental measurements for the cases (Jl =0.75m/s, Jg =0.09m/s, Jl =0.75m/s, Jg

=0.135m/s and Jl =1.0m/s, Jg =0.135m/s). However, for the other cases, the prediction

showed quite different behaviors.

(2) Based on the current experimental data, the choice of the bubble horizontal

dimension slightly modified the prediction of the void fraction.

The following characteristics are considered to be responsible of the above

observations. (1) According the developing status as analyzed in Fig. 4.9 of Chapter 4,

for the cases (Jl =0.75m/s, Jg =0.09m/s, Jl =0.75m/s, Jg =0.135m/s and Jl =1.0m/s, Jg

=0.135m/s), the flow showed better developed status at the location z=16DH. However,

for the other cases the liquid phase velocity was not either lifted up for the low void

fraction cases or entered the third developing stage for the large void fraction cases,

which deviates from the assumptions of Eq. (6.25). (2) The current bubble size is

around 3.5mm~5.5mm, among which there is little effect of DH0 or DHα on the lift

coefficients CL.

Fig. 6.15 Schematic diagram of the turbulent components along the diagonal line.

133

Fig. 6.16 Comparison between numerical predicted void fraction α and experimental measurement for Jl =0.5m/s at z=16DH:

with Jg =0.09m/s (a) along the bisector line; (b) along the diagonal line;

with Jg =0.135m/s (c) along the bisector line; (d) along the diagonal line.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Distance from the wall x, (m)

void

fracti

on

Jl=0.5 J

g=0.09

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.5 J

g=0.09

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.5 J

g=0.135

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Distance from the wall x, (m)v

oid

fracti

on

Jl=0.5 J

g=0.135

experimental data

Predicted with DH

Predicted with DH0

(a) (b)

(c) (d)

134


with Jg =0.09m/s (a) along the bisector line; (b) along the diagonal line; with Jg =0.135m/s (c) along the bisector line; (d) along the

diagonal line; with Jg =0.18m/s (e) along the bisector line; (f) along the diagonal line.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.75 J

g=0.09

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.75 J

g=0.135

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.75 J

g=0.18

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.75 J

g=0.09

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.75 J

g=0.135

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=0.75 J

g=0.18

experimental data

Predicted with DH

Predicted with DH0

(a)

(b)

(e) (c)

(d) (f)

135


with Jg =0.09m/s (a) along the bisector line; (b) along the diagonal line; with Jg =0.135m/s (c) along the bisector line; (d) along the

diagonal line; with Jg =0.18m/s (e) along the bisector line; (f) along the diagonal line.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=1.0 J

g=0.09

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=1.0 J

g=0.135

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=1.0 J

g=0.18

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=1.0 J

g=0.09

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=1.0 J

g=0.135

experimental data

Predicted with DH

Predicted with DH0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


void

fracti

on

Jl=1.0 J

g=0.18

experimental data

Predicted with DH

Predicted with DH0

(a)

(b)

(e) (c)

(d) (f)

136

6.4 Summary and Discussion

In this chapter, the void fraction effect on the bubble deformation was firstly

studied. Numerical validation of the void fraction was then carried out based on the

experimental results. The following findings were obtained:

(1) For the bubbly flow, the existence of the many bubbles could affect the

statistical bubbles’ deformation. Compared to the single-bubble case, the

bubbles behaved more prolate with the increase of the void fraction.

(2) Considering the void fraction effect, a new correlation for the statistical

bubble deformation were proposed and validated combining the bubble size

and void fraction.

(3) The numerical validation showed that only for the cases with the

better-developed flow, the predictions of the void fractions were in well

consistent with the experimental measurements.

It is notable that the analysis on the void fraction effect on the bubble

deformation was based on some assumptions such as all the ellipsoidal bubble shape

and enough bubble number during the measurements. Since the measurements were

carried out in the large duct, the wall effect was also neglected. Whether these

assumptions are suitable or not still needs further validation based on more

experimental data in the future. In addition, currently the new correlation of the bubble

deformation was proposed and validated for the mean bubble size around

3.5mm~5.5mm and the limited void fraction range. For the wider range of the bubble

size and flow conditions, it still needs further study in the future. The numerical work

further suggests the importance of knowing the developing status for the modeling of

the turbulent bubbly flow.

137

Reference

[6.1] Ishii, M., Hibiki, T., Thermo-Fluid Dynamics of Two-phase flow, Second

edition, 2010, Springer.

[6.2] Tomiyama, A., Shimada, N., 2001, A numerical method for bubbly flow

simulation based on a many-fluid model, Trans. ASME, J. of Pressure Vessel

Tech. 23, 510-516.


properties of bubbly flows, J. Fluid Mech. 495, 77–118.

[6.4] Lu, J. C., Tryggvason, G., 2008, Effect of bubble deformability in turbulent

bubbly upflow in a vertical channel, Phys. Fluids 20, 040701.

[6.5] Clift, R., Grace, J.R., Weber M.E., 1972, Bubbles, drops, and particles, Science.

[6.6] Tomiyama, A., Kataoka, I., Zun, I., and Sakaguchi T, 1998, Drag coefficients of

single air bubbles under normal and micro gravity conditions, JSME Int. J., Ser.

B 41, 472

[6.7] Magnaudet, J., Eames, I., 2000, The motion of high-Reynolds number bubbles

in homogeneous flows, Annual Rev. Fluid Mech. 32, 659-708.



[6.9] Legendre, D., 2007, On the relation between the drag and the vorticity produced

on a clean bubble, Phys. Fluids 19, 018102.

[6.10] Rastello, M., Mari´e, J. L., Lance, M., 2011, Drag and lift forces on clean

spherical and ellipsoidal bubbles in a solid body rotating flow, J. Fluid Mech.

682, 434.

[6.11] Serizawa, A., Kataoka, I., Michiyoshi, I., 1975, Turbulence structures of

air-water bubbly flow-I. measuring techniques, Int. J. Multiphase Flow 2,

221-233.

[6.12] Liu, T. J., 1993, Bubble size and entrance effects on void development in a

vertical channel, Int. J. Multiphase Flow 19, 99-113.

138

[6.13] Liu, T. G., Bankoff, S. G., 1993, Structure of air-water bubbly flow in a vertical

pipe—II. Void fraction, bubble velocity and bubble size distribution, Int. J. Heat

and Mass Transfer 36, 1061–1072.

[6.14] Fujiwara, A., Danmoto, Y., Hishida, K., Maeda, M., 2004, Bubble deformation

and flow structure measured by double shadow images and PIV/LIF, Exp.

Fluids 36, 157–165.

[6.15] Sun, H. M., 2014, Study on upward air-water two-phase turbulent flow

characteristics in a vertical large square duct, Doctor Thesis, Kyoto University.

[6.16] Clark, N. N., Turton R, 1988, Chord length distributions related to bubble size

distributions in manyphase flows, Int. J. Multiphase Flow 14, 413-424.

[6.17] Liu, W., Clark, N. N., 1995, Relationships between distributions of chord length

and distributions of bubble sizes including their statistical parameters, Int. J.


[6.18] Wellek, R. M., Agrawal, A. K., Skelland, A, H, P, 1966, Shape of liquid drops

moving in liquid media, AIChE J. 12, 854-862.

[6.19] Tomiyama, A., Celata, G. P., Hosokawa, S., Yoshida, S., 2002, Terminal

velocity of single bubbles in surface tension force dominant regime, Int. J.


[6.20] Ishii, M., Zuber, N., 1979, Drag coefficient and relative velocity in bubbly,

droplet or particulate flows, AIChE J. 25, 843-855.

[6.21] Garnier, C., Lance, M., Mariém J. L., 2002, Measurement of local flow

characteristics in buoyancy-driven bubbly flow at high void fraction,

Experimental Therm. Fluid Sci. 26, 811-815.



Flow 13, 327–343.

139

Chapter 7

Turbulence modulation of the turbulent bubbly flow

7.1 Introduction

As compared with the single-phase turbulent flow, for the turbulent bubbly flow

the presence of the bubbles will modulate the turbulence characteristics by the following

mechanisms: (1) Altering the shear-induced turbulence by modifying mean flow

properties such as the mean shear rate; (2) Inducing additional pseudo turbulence by

agitating the flow such as wake structure behind the bubbles; (3) Interacting with the

existing shear-induced eddies by being captured or colliding; (4) Inducing additional

interaction between bubble-induced and shear-induced eddies. Due to the above

mechanisms, the turbulence modulation such as turbulence enhancement or suppression

occurs in the turbulent bubbly flow, which has been partially discussed in [7.1].

Though several studies on the turbulence modulation [7.2-7.13] have been

carried out for the turbulent bubbly flow, the understanding and modeling are still

unsatisfactory so far. In this Chapter, the turbulence modulation of the upward turbulent

bubbly flow will be further studied based on the recently established experimental

database. It will aim at understanding the following questions:

(1) How much turbulent energy was induced by the bubbles?

(2) How does the interaction between the bubble-induced turbulence and the

wall-induced turbulence behave?

(3) What is the property of the bubble-induced turbulence such as the

turbulence anisotropy?

(4) How to model the bubble induced turbulence considering the above

properties?

140

7.2 Turbulent kinetic energy

For the turbulent bubbly flow in the ducts, the turbulence generation includes the

wall-induced turbulence tkw which is the main mechanism for the single-phase turbulent

flow and the bubble-induced turbulence tkb. Several studies [7.3-7.4, 7.8-7.11] have

been conducted to model the bubble-induced turbulence by comparing the turbulence

with and without the bubbles. The theoretical analysis could be found in [7.4, 7.11],

which estimated the bubble-induced turbulence based on the potential theory [7.4] and

based on the mean flow velocity and the wall shear stress [7.11], respectively. The

uniform bubbly flow experiments in [7.9] showed that below the critical void fraction αc

of the order of 1%, the bubble-induced pseudo turbulence could be well estimated by

the potential flow model, and above the critical void fraction αc, the turbulence was

strongly amplified by the hydrodynamic interactions between the bubbles and also the

background turbulence. The DNS study [7.14] showed a large difference from the

prediction by potential flow model.

This part focuses on the question of how much turbulent energy could be

induced by the bubbles. The turbulence databases shown in Chapter 2 will be adopted.

For the turbulent bubbly flow in the ducts, the wall-induced turbulence tkw and the

bubble-induced turbulence tkb are coupled with each other. Therefore, the knowledge of

both the solely wall-induced turbulence and the solely bubble-induced turbulence are

required to understand the final turbulent characteristics therein. For the solely

wall-induced turbulence, numerous researchers has been carried out for the single-phase

flow, based on which we reached a comprehensive understanding. However, for the

solely bubble-induced turbulence, there are still very few studies except the experiments

by Lance and Bataille about the uniform bubbly flow for the lower void fractions lower

than 3% [7.9]. For cases with larger void fraction, the solely bubble-induced turbulence

is different due to the stronger effect of the bubble-bubble interaction. To quantify the

solely bubble-induced turbulence for a wider range of void fraction, the recently

established experimental database of the bubbly flow in the ducts will be used and

141

analyzed in the following. Firstly the comparison between the bubble-induced

turbulence tkb and wall-induced turbulence tkw will be carried out to understand the role

of the bubble-induced turbulence. After that, the relations between the bubble-induced

turbulence tkb, the void fraction α and wall-induced turbulence tkw will be considered.

Finally, the interaction between bubble-induced turbulence and wall induced turbulence

will be discussed based on their relations.

The role of the bubble-induced turbulence here is considered by the ratio of the

turbulent kinetic energies between the turbulent bubbly flow tkt and the single-phase

flow tks under the same flow rate according to existed data, which were shown in Figs.

7.1-7.5 for the circular pipes with small and large diameter and also the noncircular

ducts. Larger ratio tkt / tks indicates stronger role of the bubble-induced turbulence. Here,

generally two-sets of the cases with closer liquid flow rates are given for simplicity and

comparison. Similar tendency could be found for the other cases. As shown from Figs.

7.1-7.5, the following trends could be obtained:

(i) Void fraction and liquid flow rate effect

The ratio tkt / tks increases with the increase of the void fraction for the same

liquid flow rate, and shows larger for cases with the smaller liquid flow rate for the

similar void fraction. It is easy to understand this observation because of the increase of

the bubble-induced turbulence with increasing the void fraction and the decrease of the

wall-induced turbulence with decreasing the liquid flow rate.

(ii) Turbulence enhancement or reduction

Generally, the ratio tkt / tks is greater than 1 especially in the duct center of the

large duct indicating the turbulence enhancement by the bubbles, except several cases in

the small circular pipe with very small gas flow rate and high liquid flow rate where the

turbulence reduction could be observed in the near wall region such as Jl=1.03m/s and

X%=0.0085 in Serizawa’s experimental database [7.2-7.3]. The turbulence reduction

under these cases was discussed in [7.8] and will be considered later in this chapter.

142

(iii) Lateral distribution

Except two cases under Jl=0.491m/s in Hibiki’s experimental database [7.15]

which were in the transition regime, generally the ratio tkt / tks for the turbulent bubbly

flow decrease from the duct center to the wall even with much higher void fraction near

the wall. It indicates the bubble-induced turbulence played a dominated role in the duct

center as compared with the near wall region. The exception of the two cases mentioned

above might be attributed to the bubble coalescence and breakup which induced another

mechanism for the turbulence generation and will not be considered here.

(iv) Geometry effect

Fig. 7.6(a) summarizes the ratio <ww>tc/<ww>sc of the axial turbulent intensity

for the turbulent bubbly flow in the duct center <ww>tc to that of the single-phase

turbulence <ww>sc for different flow conditions and ducts. It shows the larger ratio

tkt/tks could be observed for the larger duct size DH. Fig. 7.6(b) shows the contour lines

for the flow conditions Jl and Jg under which the ratio <ww>tc/<ww>sc is around 5 and 25,

respectively. Here the flow conditions on the contour lines were summarized and

interpolated from the corresponding references for different ducts. It was found that the

contour lines for the small circular pipes from Hibiki’s database [7.15] lie in the upper

region of that for the large ducts from Shakwat’s [7.13] and Sun’s database [7.16]. For

the Serizawa’s database [7.2] in the small circular pipe, the critical conditions for

<ww>tc/<ww>sc ≈ 25 were not reached even in the slug regime. It means that under the

same flow condition, the ratio <ww>tc/<ww>sc in the larger ducts is larger than that in the

small ducts. For example, for the case of Jl ≈1.0m/s and Jg≈0.1m/s as shown in

Fig.7.6(b), <ww>tc/<ww>sc ≈10 in the large ducts while <ww>tc/<ww>sc ≈5 in the small

ducts. Therefore, the bubble-induced turbulence played a more dominate role in the

larger ducts especially in the core region. The following two factors might be

responsible for this observation (a) more space for the bubble’s horizontal deformation

and wake development behind the bubbles for the large ducts which induced stronger

turbulence; (b) less effect from the wall-induced turbulence for the large ducts.

143

Fig. 7.1 Lateral distributions of the ratio of the turbulent kinetic energies between tkt and

tks in the small circular pipe with DH=60mm based on Serizawa’s experimental database

under (a) Jl =0.74m/s and (b) Jl =1.03m/s [7.2].


tks in the small circular pipe with DH=50.8mm based on Hibiki’s experimental database:


0

5

10

15

r/R

t kt/t

ks

Jl=0.74m/s X %

0.000

0.0119

0.024

0.0358

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

0

5

10

15

r/R

t kt/t

ks

Jl=1.03m/s

X %0.000

0.0085

0.0170

0.0258

0.0341

0.0427

0.0511

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

20

40

60

80

r/R

t kt/t

ks

J

l=0.491m/s J

g

0.000

0.0275

0.0556

0.129

0.190

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

10

20

30

40

r/R

t kt/t

ks

J

l=0.986m/s J

g

0.000

0.0473

0.113

0.242

0.321

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

(a) (b)

(a) (b)

144


tks in the small circular pipe with DH=60mm based on Liu’s experimental database [7.7].


tks in the large circular pipe with DH=200mm based on Shakwat’s experimental database


0 0.2 0.4 0.6 0.8 10

20

40

60

r/R

t kt/t

ks

J

l J

g

0.376 0.00

0.376 0.18

0.753 0.18

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

r/R

0

20

40

60

80

r/R

t kt/t

ks

Jl=0.45m/s J

g

0.0000.015

0.0850.100

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

r/R

0

20

40

60

r/R

t kt/t

ks

J

l=0.68m/s J

g

0.0000.015

0.0500.085

0.1000.180

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

r/R

(a) (b)

145

Fig. 7.5 Lateral distributions of the ratio of the turbulent kinetic energies between tkt and tks in the large circular pipe with DH=136mm

based on Sun’s experimental database: under Jl =0.75m/s (a) along the bisector line; (b) along the diagonal line;

under Jl =1.0m/s (c) along the bisector line; (b) along the diagonal line [7.16].

0

20

40

60

t kt/t

ks

Jl=0.75m/s Bisector

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

2x/DH

0

20

40

60

t kt/t

ks

J

l=0.75m/s Diagonal

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

2x/DH

0

10

20

30

t kt/t

ks

Jl=1.0m/s Bisector

0 0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

2x/DH

0.0000.0900.135

0.1800.225

0

10

20

30

t kt/t

ks

J

l=1.0m/s Diagonal

0.0000.0900.135

0.1800.225

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

2x/DH

(a) (b)

(c) (d)

Jg Jg

Jg

Jg

146

Fig. 7.6 (a) The relation between the ratio <ww>tc/<ww>sc and the local void fraction α in

the duct center for different ducts; (b) The contour lines for the flow conditions Jl and Jg

when <ww>tc/<ww>sc ≈ 5, 10, 25.

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

c

<w

w>

tc /

<w

w>

sc

Liu

Serizawa

Sun

Shawkat

(a)

(b)

147

7.2.1 Bubble-induced turbulence at the duct center

Based on the above analysis, in the following the turbulent intensities in the duct

center will be mainly considered and analyzed especially in the large ducts where the

bubble-induced turbulence dominates the local turbulence in order to quantify and

model the bubble-induced turbulence. Here, we consider the turbulence in the duct

center as the solely bubble-induced turbulence since therein the shear-induced

turbulence is very weak due to the flat liquid phase velocity profile or small liquid phase

velocity shear. In addition, we also neglect the horizontal diffusion and convective

effect which are extremely weak in the core region. In other words, the turbulence in the

core region was mainly induced because of the bubble agitation.

Fig. 7.7 shows the relation between the axial turbulent intensities <ww>c and the

local void fraction α in the duct center based on the above experimental databases.

Besides, three correlations for the bubble-induced turbulence were also shown in Fig.

7.7, which are 22.0 rW proposed by [7.9, 7.11] based on the potential theory and

experimental results for α < 1%, 2

rW proposed by [7.14] by fitting their DNS results

for α<24%, and 6.172.0 proposed by [7.5] by fitting their experimental results

including the near wall region for α>1.3%. Wr is the relative velocity between the

bubbles and liquid phase and Wr=0.23m/s (the terminal velocity) in the above

correlations. As shown from Fig. 7.7(a), the following trends could be obtained:

(i) Duct geometry effect

Under the same void fraction, the bubble-induced turbulence is stronger in the

large ducts as compared to that in the small circular pipes. The discrepancy increases

with the increase of the void fraction. Similarly it could also be attributed to the

suppression of the bubble’s horizontal deformation and wake development in the small

circular pipes.

(ii) Liquid flow rate effect

148

The turbulence for the turbulent bubbly flow in the large ducts shows much

more scattered than that in the small circular pipes. The scattering in the large ducts was

found to be related with different liquid flow rates, which is consistent with [7.9].

Fig.7.7 (b) shows the ratio <ww>/Wl for the above cases. It was found that after dividing

by the local liquid velocity Wl or Reynolds number Rel, the scattering could be

significantly improved especially in the large duct. In addition, based on <ww>/Wl the

dependence of the turbulence on the duct size could be clearly shown especially for the

cases with α>5%. The deviation for further higher void fraction could be attributed to

that the flow was in the transition regime where bubble-coalescence is needed to be

considered. The ratio <ww>/Wl shows linear almost linear increase with the void

fraction α, where the slope should be determined by the duct size DH, the bubble size DB,

the bubble deformation E, and the relative velocity of the bubbles Wr. However, how it

relates to these parameters, it still needs further study in the future.

(iii) Correlation validation of 22.0 rW

Generally, the bubble-induced turbulence is larger than the predictions by

22.0 rW especially in the large ducts, which indicates the important role of the wake

behind the bubbles to the turbulence generation. However, in the small circular pipe, the

discrepancy between the potential theory and experimental measurements is apparently

smaller than that in the larger ducts. It is noticed that in the two-fluid models proposed

by Lahey [7.17], Bertodano et al. [7.18], Politano et al. [7.19], and Hosokawa and

Tomiyama [7.20], the bubble-induced turbulence was predicted by this potential theory

and validated based on the experimental database in the small ducts. The predictions

therein were basically in consistent with the experimental measurements. However, for

the large pipes the prediction shows larger discrepancy, and the discrepancy increases

with the void fraction α. Therefore, the new model considering the bubble-induced

wake and bubble deformation is required for the turbulent bubbly flow in the large ducts

for more accurate turbulence prediction therein.

149

(iv) Correlation validation of 2

rW

The turbulence because of the wake behind bubbles was included in the

correlation 2

rW [7.14] by fitting their DNS results with tens of the bubbles in a

limited domain. The prediction shows larger than the experimental measurements in the

small circular pipes but still smaller than that in the larger circular and noncircular ducts,

which might be attributed to the limited bubble number and computational domain.

(v) Correlation validation of 6.172.0

The correlation 6.172.0 was proposed in [7.5] by directly extracting the

wall-induced single-phase turbulence from the total turbulence of the turbulent bubbly

flow even the whole flow region. Though the prediction shows closer to the

experimental results in the large ducts, it is lack of the physical considerations and the

scattering phenomenon of the turbulence for different liquid flow rates could not be

predicted.

In addition, all the above correlations are independent of the bubble size and

deformation which played important roles in the bubble-induced turbulence. It made the

predictions deviate more from the experimental measurements in the large ducts.

In [7.21], based on the plausible assumption that all the energy lost because of

the drag resistance drag

LF on the bubbles was converted to the turbulent kinetic energy

in the bubble wake, the additional source term B

LS in the liquid phase turbulent kinetic

energy tk transport equation was proposed for the bubbles contribution as follows:

rL

B

LS VF drag. (7.1)

However, it is notable that for the turbulent bubbly flow the work done by the

drag force drag

LF contributes not only to the turbulent kinetic energy tk but also the

mean kinetic energy tM, where the total energy in the flow is kMtotal ttt . Hence, in

this way the double computation of the mean energy occurs in the governing equations

150

of the bubbly flow. This was mentioned and discussed by Yokomine et al. for the

turbulent flow with the dispersed solid particles [7.22]. Therein the double computation

was extracted from the source term by modifying the drag coefficient as follows:

StokesDD

W

D CCC , (7.2)

where StokesDC , is the drag coefficient for the dispersed particles based on the Stokes

flow.

In the following, this method proposed by Yokomine et al. [7.22] will be

adopted. For simplicity, the energy balance for the liquid phase turbulence kinetic

energy only at the duct center will only be considered where the shear contribution, and

horizontal diffusion and convection could be neglected. Therein, the turbulent kinetic

energy tkc was determined by the local bubble contribution B

LS and dissipation rate L

as follows.

0drag LrLL

B

LS VF . (7.3)

For the turbulent bubbly flow, the dissipation rate L is not only related with

the eddy dissipation due to viscosity similar to the single-phase turbulent flow but

also the dispersed bubbles B due to the bubble deformation, the capture of the

bubbles by eddies and also the collisions between the bubbles and eddies. The

contribution of the dispersed particles to the dissipation rate L is related with the

characteristic length and time of both the dispersed bubbles and eddies, which was

discussed in [7.22]. However, the case with the large light particles such as large

bubbles was not considered in detail because of its complexity such as the deformability,

which still needs further studies in the future. Here, as an initial step the dissipation rate

L will be considered based on the effective viscosity t in which for the turbulent

bubbly flow the bubble contribution is needed to be considered:

2

kL

t

tC

(7.4)

151

where Cμ is the constant and around 0.09 for the single-phase turbulent flow.

Substituting Eq. (7.4) into Eq. (7.3), it is obtained that

rLt

kC

t VF drag

. (7.5)

At the duct center, the eddy viscosity t induced by the bubbles is considered

by the model proposed by Sato and Sadatomi [7.23],

rBbt wDC (7.6)

Substituting Eqs. (1.4), (7.2) and (7.6) into Eq. (7.5), the turbulent kinetic energy

tk in the liquid phase could be determined as follows:

b

StokesDD

rk CC

CCWt

,2

(7.7)

where StokesDC , was determined by ,Stokes 24 / ReDC ; Cμ is chosen to be 0.09

similar to the single-phase turbulent flow; Cμb is chosen to be 0.6; the correlation

proposed by Tomiyama et al. [7.24] was used to determine drag coefficients as

5.0

687.0

1

1

4Eo

Eo

3

8,

Re

72,Re15.01

Re

24minmax

B

B

B

DC .

Fig. 7.8 shows the turbulent kinetic energy tk based on Eq. (7.7) for different

bubble diameters, from which better prediction of tk could be obtained for the

turbulent bubbly flow in the large ducts. Hence, Eq. (7.7) is suggested for the

turbulent bubbly flow in the large ducts. However, the effect of the bubble

deformation still needs further consideration in the future.

152

Fig. 7.7 (a) The axial turbulent intensities <ww> vs. the local void fraction α in the duct

center; (b) <ww>/Wl vs. the local void fraction α in the duct center.

0 0.05 0.1 0.15 0.2 0.25 0.30

0.02

0.04

0.06

0.08

<w

w>

Serizawa

Sun

Shawkat

0.2 Wr

2

Wr

2

0.721.6

0 0.05 0.1 0.15 0.2 0.25 0.30

0.02

0.04

0.06

0.08

<w

w>

/Wl

Serizawa

Sun

Shawkat

0.35

0.15

0.02

(a)

(b)

153

Fig. 7.8 Comparison the experimental data of the turbulent kinetic energy tk of the liquid

phase with the new correlation Eq. (7.7) for different bubble diameter at the duct center.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

t k

Liu

Serizawa

Sun

Shawkat

0.5 Wr

2

DB

=3mm

DB

=4mm

DB

=5mm

DB

=6mm

154

7.2.2 Bubble-induced and wall-induced turbulence interaction

In the near wall region, the turbulence in the liquid phase includes the

wall-induced and bubble-induced. To investigate the bubble-induced turbulence, the

wall-induced contribution is necessary to be extracted. However, near the wall the

wall-induced turbulence could be strongly modified by the bubbles, thus it might be not

suitable to directly extract the wall-induced single-phase turbulence from the total

turbulence of the turbulent bubbly flow. In other words, it is very difficult to directly

investigate the bubble-induced turbulence in the near wall region. In the present study,

the bubble-induced turbulence and wall-induced turbulence will be investigated near the

wall in an indirect way hereafter.

7.2.2.1 Wall effect on bubble-induced turbulence

As discussed in Section 7.2.1, the turbulence in the duct center could be

considered solely induced by bubbles. Firstly, the ratio of the local turbulent kinetic

energy tkt to that in the duct center c

ktt will be analyzed and shown in Figs. 7.9-7.13 for

the circular pipes with small and large diameter and also the noncircular ducts based on

the existed experimental databases. The following trends could be observed:

(1) From Figs. 7.9-7.13, except two cases under Jl=0.491m/s from Hibiki’s

experimental databases [7.15] in the transition regime, generally the ratio tkt /

c

ktt for

the single-phase turbulent flow is much larger than that for the turbulent bubbly flow

even with higher void fraction near the wall, i.e.

cB

kb

wB

kb

wB

kw

cB

kb

cB

kw

wB

kb

wB

kw

cS

kw

wS

kw

t

tt

tt

tt

t

t,

,,

,,

,,

,

,

. (7.8)

where wS

kwt , and

cS

kwt , are the wall-induced turbulence near the wall and in the duct

center for the single-phase turbulent flow; wB

kwt , and

cB

kwt , are the wall-induced

turbulence near the wall and in the duct center for the turbulent bubbly flow; wB

kbt , and

cB

kbt , are the bubble-induced turbulence near the wall and in the duct center for the

155

turbulent bubbly flow. The exceptions in the small circular pipes mentioned above

might be attributed to the bubble coalescence and breakup which induced another

mechanism for the turbulence generation, which will not be considered here.

Rearranging Eq. (7.7), it could be obtained that

cS

kw

cB

kb

wS

kw

wB

kw

t

t

t

t,

,

,

,

, (7.9)

indicating that for the turbulent bubbly flow the modification of the wall-induced

turbulence is much less than that in the duct center. In addition, from Eq. (7.8) we also

have

cB

kb

wB

kb

cB

kb

wB

kb

wB

kw

t

t

t

tt,

,

,

,,

. (7.10)

Based on Eq. (7.10), the wall effect on the bubble-induced turbulence could be

considered. Due to the non-uniform void fraction distribution, we need to know how

much is the bubble-induced turbulence in the absence of the wall before in order to

consider the wall effect on the bubble-induced turbulence, which will be considered by

the proposed correlation Eq. (7.8) and the linear relation with the void fraction, i.e.,

c

w

cB

kb

wB

kb

t

t

,

,

. (7.11)

Substituting Eq. (7.11) to Eq. (7.10), it should be

c

w

cB

kb

wB

kb

wB

kw

t

tt

,

,,

. (7.12)

Fig. 7.14 shows the comparison between the left and right hands of Eq. (7.11)

based on Sun’s experimental database. In contrast, it is observed that

c

w

cB

kb

wB

kb

wB

kw

cB

kb

wB

kb

t

tt

t

t

,

,,

,

,

, (7.13)

indicating that comparing to the core region under the same void fraction the

156

bubble-induced turbulence is reduced by the wall effect. The following two mechanisms

could be considered to be responsible for this observation: (a) the wall effect elongates

the bubble shape; (b) the wall suppressed the bubble wake development.

The scattering in Fig. 7.14 is due to the different liquid flow rates, i.e.,

l

c

w

cB

kb

wB

kb

wB

kw JFt

tt,

,

,,

. (7.14)

Considering / ,w c l cF J H as shown in Fig. 7.15(a), Fig. 7.15(b)

shows comparison between , , ,/B w B w B c

kw kb kbt t t and c , in which the scattering was

significantly improved. Besides, it was found that , , ,/B w B w B c

kw kb kbt t t linearly decreases

with the increase of c , indicating the turbulence near the wall could be predicted by the

following correlation:

c

cB

kb

wB

kb Htt ,, . (7.15)

cH shows the suppression of the bubble-induced turbulence and is related

with duct geometry and the local void fraction. It still needs further researches and

verification for different cases in the future.

From Figs. 7.9-7.13, it was also observed that for the single-phase turbulent flow

the ratio tkt/c

ktt is always larger than 1, since the turbulence is mainly generated near the

wall and decayed to the duct center. However, it is not always true for the turbulent

bubbly flow even for the cases with higher void fraction near the wall, i.e.,

cB

kb

wB

kb

wB

kw

cB

kb

cB

kw

wB

kb

wB

kw

cS

kw

wS

kw

t

tt

tt

tt

t

t,

,,

,,

,,

,

,

1

or w

wB

kb

wB

kwc

cB

kb ttt ,,, . (7.16)

For the turbulent bubbly flow in the large ducts as shown in Figs. 7.12-7.13

where the void fraction shows a flatter distribution, this tendency could be obviously

observed and further confirms wall suppression of the bubble-induced turbulence. In the

small circular pipe, increasing the void fraction when the near wall turbulence was

dominated by the bubbles, this phenomenon could be observed as shown in Fig. 7.11.

157

Fig. 7.9 Lateral distributions of the ratio of the local turbulent kinetic energy tkt to that in

the duct center c

ktt

in the small circular pipe with DH=60mm based on Serizawa’s


Fig. 7.10 Lateral distributions of the ratio of the local turbulent kinetic energy tkt to that

in the duct center c

ktt

in the small circular pipe with DH=50.8mm under (a) Jl

=0.491m/s and (b) Jl =0.986m/s based on Hibiki’s experimental database [7.15].

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

r/R

t k/t

c k

Jl=0.74m/s X %

0.000

0.0119

0.024

0.0358

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

r/R

t k/t

c k

Jl=1.03m/s X %

0.000

0.0085

0.0170

0.0258

0.0341

0.0427

0.0511

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

r/R

t kt/t

c kt

Jl=0.491m/s J

g

0.000

0.0275

0.0556

0.129

0.190

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

r/R

t kt/t

c kt

Jl=0.986m/s J

g

0.000

0.0473

0.113

0.242

0.321

(a) (b)

(a) (b)

158


in the duct center c

ktt

in the small circular pipe with DH=60mm based on Liu’s



in the duct center

c

ktt in the large circular pipe with DH=200mm under (a) Jl =0.45m/s

and (b) Jl =0.68m/s based on Shakwat’s experimental database [7.13].

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

r/R

t kt/t

c kt

Jl J

g

0.376 0.00

0.753 0.00

0.376 0.18

0.535 0.18

0.753 0.18

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

r/R

t kt/t

c kt

Jl=0.45m/s J

g

0.0000.0150.030

0.0650.0850.100

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

r/R

t kt/t

c kt

Jl=0.68m/s J

g

0.0000.0150.0500.065

0.0850.1000.180

(a) (b)

159

Fig. 7.13 Lateral distributions of the ratio of the local turbulent kinetic energy tkt to that in the duct center

c

ktt in the large noncircular

pipe with DH=136mm based on Sun’s experimental database: under Jl =0.75m/s (a) along the bisector line; (b) along the diagonal line;

under Jl =1.0m/s (c) along the bisector line; (b) along the diagonal line [7.16].

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

t kt /

t kt

c

Jl=0.75m/s Bisector

Jg

2x/DH

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

t kt /

t kt

c

Jl=0.75m/s Diagonal

Jg

2x/DH

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

t kt /

t kt

c

Jl=1.0m/s Bisector

Jg

2x/DH

0.0000.0900.135

0.1800.225

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

t kt /

t kt

c

Jl=1.0m/s Diagonal

Jg

2x/DH

0.0000.0900.135

0.1800.225

(a) (b)

(c) (d)

160

Fig. 7.14 Comparison between , , ,/B w B w B c

kw kb kbt t t and /w c based on Sun’s


1 2 3 4 5 60

1

2

3

4

5

6

w

/c

(tk

w

B,w

+t k

b

B,w

) /t

kb

B,c

Jl=0.50

Jl=0.75

Jl=1.00

Jl=1.25

(tkw

B,w+t

kb

B,w) /t

kb

B,c=

w /

c

161

Fig. 7.15 (a) Relation between /w c and c ; (b) Relation between

, , ,/B w B w B c

kw kb kbt t t and c based on Sun’s experimental database [7.16].

0 0.05 0.1 0.15 0.21

2

3

4

5

6

w

/

c

Jl=0.50

Jl=0.75

Jl=1.00

Jl=1.25

0 0.05 0.1 0.15 0.20

2

4

6

8

10

c

(tk

w

B,w

+t k

b

B,w

) /t

kb

B,c

Jl=0.50

Jl=0.75

Jl=1.00

Jl=1.25

(a)

(b)

162

7.2.2.2 Bubble effect on wall-induced turbulence

It is difficult to consider the bubble effect on wall-induced turbulence near the

wall coupling with the modification of the bubble-induced turbulence by the wall effect.

As compared to the single-phase turbulent flow, generally the turbulence in the liquid

phase increases with the void fraction for the turbulent bubbly flow, the tendency of

which is shown in Fig. 7.16. For cases with low void fraction, the turbulence reduction

in the liquid phase was found as compared to the single-phase turbulent flow [7.8].

Fig.7.17 shows the experimental conditions under which turbulence reduction in the

liquid phase was observed. Hereafter, a simple criterion for turbulent reduction will be

considered in the turbulent bubbly flow.

For the case with the turbulence reduction in the liquid phase, we have

wS

kww

wB

kb

wB

kw ttt ,,, . (7.17)

As for the bubble-induced turbulence, though it was suppressed by the wall

effect, it should be stronger than that the pseudo-turbulence based on the potential

theory, i.e.,

2,

2

1Rww

wB

kb Wt . (7.18)

Substituting Eq. (7.18) to Eq. (7.17), we have

2,,

2

1Rw

wS

kw

wB

kw Wtt . (7.19)

Based on Eq. (7.19), the critical void fraction near the wall could be obtained as

2, /2 R

wS

kwcalc Wt . It indicates that the turbulent enhancement will occur if

2, /2 R

wS

kww Wt and the turbulent reduction is only possible to occur if 2, /2 R

wS

kww Wt .

Fig.7.18 shows comparison the experimental measured void fraction and the calculated

void fraction based on 2, /2 R

wS

kwcalc Wt near the wall for the cases including

turbulence enhancement and reduction. Since it is lack of the experimental database of

163

the void fraction and near wall turbulence, only the data from Shakwat et al. [7.13] and

Sun [7.16] were shown, based on which the above consideration was confirmed.

Moreover, it explains the reason why the turbulence reduction could occur under higher

void fraction for larger liquid flow rate but lower void fraction for smaller liquid flow

rate.

As for the mechanism for the turbulence reduction near the wall under certain

conditions, in [7.8] it was attributed to the following two effects due to the bubbles: (1)

large velocity fluctuations gradient near the bubble interface which increases the

turbulent energy dissipation; (2) energy dumping effect due to the bubble deformation.

Here, another two mechanisms responsible for the reduction of the wall-induced

turbulence in the turbulent bubbly flow are proposed as shown in Fig. 7.19.

(i) Flow acceleration laminarization

As shown in Fig. 7.19(a), due to the formation of the bubble layer near the wall,

the liquid flow was confined between the wall and the bubble layer, which laminarized

the flow due to the local flow acceleration due to the bubbles. However, it still needs

further experimental measurement including the velocity and turbulence in the inner

layer to confirm the above points.

(ii) Suppression of “bursting event” of the coherent structures

For the single-phase flow, the turbulence generation is mainly due to the

―bursting event‖ of the coherent structures near the wall as shown in shown in Fig.

7.19(b). However, the presence of the bubbles whose size is on the similar order of the

coherent structures broke these structures, thus suppressed the source of the

wall-induced turbulence.

164

Fig. 7.16 Schematic diagram of the relation between the turbulence in the liquid phase

and the void fraction for the turbulent bubbly flow.

Fig. 7.17 Experimental conditions with turbulence suppression and enhancement in the

liquid phase from [7.8].

10-3

10-2

10-1

100

10-1

100

101

Jg m/s

Jl m

/s

Serizawa [7.2,7.8]

Inoue [7.25]

Sullivan [7.26]

Ohba [7.27]

Aoki [7.28]

Lee et [7.29]

Shakwat [7.13]

Sun [7.16]

Turbulence Reduction

Turbulence Enhancement

165

Fig. 7.18 Comparison the experimental measured void fraction and the calculated void

fraction based on 2, /2 R

wS

kwcalc Wt near the wall.

Fig. 7.19 Schematic diagram of the mechanism for the turbulence suppression.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

calc

ex

p

Shakwat [7.13] turbulence reduction

Sun [7.16] turbulence enhancement

Shakwat [7.13] turbulence enhancement

Turbulence enhancement

occurs.

Turbulence reduction is

possible to occur.

166

7.3 Turbulence anisotropy in turbulent bubbly flow

As mentioned in Chapter 1, so far there are very few detailed discussions on the

turbulence anisotropy of the turbulent bubbly flow, based on which the qualitative

effects of the bubbles on the turbulence anisotropy are still contradictory [7.6, 7.9,

7.30-7.31]. Therefore, how the bubbles affect turbulence anisotropy and how to model

turbulence anisotropy are still two open questions. This section is aimed at these two

topics. Firstly, the bubble effects on turbulence anisotropy will be analyzed in the liquid

phase based on the currently established database for bubbly flows in circular and

noncircular ducts. According to the analysis, the suggestions on the future model of the

turbulence anisotropy will be given then.

7.3.1 Bubble effects on turbulence anisotropy in liquid phase

Firstly, the turbulence anisotropy tensor b was defined to show the strength of

turbulence anisotropy as follows:

/ 2 1 / 3 / 2 / 2

/ 2 / 2 1 / 3 / 2

/ 2 / 2 / 2 1 / 3

k k k

k k k

k k k

uu t uv t uw t

uv t vv t vw t

uw t vw t ww t

b . (7.20)

where b11+ b22+ b33=0; for the isotropic turbulence, b=0. Here, the normal

components b11, b22, and b33 will be focused on. The deviation from 0 shows the

strength of the turbulence anisotropy.

Figs. 7.20-7.24 show b11, b22, and b33 calculated based on the current database

for the circular pipes with small and large diameter and the noncircular ducts. The

following trends could be observed:

(1) For the single-phase turbulent flow, the turbulence anisotropy increase

from the duct center to the near wall region, due to the high velocity

gradient and the wall suppression on the turbulent wall-normal

167

components in the near wall region.

(2) For the turbulent bubbly flow, as shown in Fig. 7.20 and Fig. 7.23,

Serizawa’s experiments [7.2] in the smaller circular pipes and Shakwat’s

experiments [7.13] with lower gas flow rates in the large circular pipe

showed the introduction of the bubbles didn’t alter the turbulent isotropy

or made the turbulence more isotropic in the whole cross section.

However, as shown in Fig. 7.21 and 7.22, experiments of Wang et al.

[7.6] and Liu and Bankoff [7.7] in the smaller circular pipes showed that

the presence of the bubbles made the turbulence more anisotropic in the

core region but more isotropic in the near wall region. Similar results

were also obtained for the turbulent bubbly flow in the large circular pipe

based on Shakwat’s experiments [7.13] with the higher gas flow rates and

in the large non-circular duct based on Sun’s experiments [7.16].

(3) For the cases as shown in Figs. 7.21-7.24 in which bubbles increased the

turbulence anisotropy, in the core region the anisotropy tensor was found

to reach a constant value to some extent. For example 22.011 b and

11.033 b where b11 and b33 are in the axial and lateral direction,

respectively.

In order to understand the above observations and model the bubble effects on

the turbulence anisotropy in liquid phase, it is necessary to discuss the following

parameters’ effects due to their crucial importance to the bubble-induced turbulence

including: (a) void faction; (b) duct geometry; (c) bubble size; (d) bubble relative

velocity; (e) bubble deformation.

Firstly, the void fraction effect is considered in the duct center, where the

turbulence might be dominated by bubbles. Fig.7.25 shows the relation between the

turbulence anisotropy tensor b11 and b33 and the local void fraction α in the duct center

168

for different ducts. Based on the potential theory [7.11], the anisotropy tensor of the

bubble-induced turbulence is as follows and shown in Fig. 7.25 as a reference:

1/ 30 0 0

0 1/ 30 0

0 0 1/15

b . (7.21)

For the turbulent bubbly flow in the large ducts based on Shakwat’s [7.13] and

Sun’s [7.16] experiments, it could be obtained that the turbulence is more isotropic for

the small void say less than 5% and the anisotropy tensor b was closer to the value

predicted based on the potential theory in Eq. (7.20), which is in consistent with that

observed by Lance [7.9]. For the higher void fraction, the turbulence is anisotropic and

the anisotropy tensor b reaches a constant value independent of the void fraction.

For the turbulent bubbly flow in the small circular pipes, Serizawa’s data

showed more isotropic turbulence closer to the potential theory and was focused in the

smaller void fraction range say less than 10%. However, Wang’s [7.6] and Liu’s [7.7]

data in the small circular pipes showed more anisotropic turbulence similar to that in the

large ducts, which were focused in the larger void fraction range larger than 10%.

Therefore, how the bubble affects the turbulence anisotropy is related with the

void fraction, and the turbulence anisotropy increase with the void fraction to a constant

value. The tendency of the void fraction effect on the turbulence anisotropic tensor b

obtained here is inconsistent with that in Trygovasson’s DNS [7.31] results for the

solely bubble-induced turbulence. In [7.31], it showed the turbulence behaves more

isotropic with increase of the void fraction by comparing the results for 2%, 6%, and

12%, which might be due to the reduction of the relative velocity with increase of void

fraction.

In addition, how the bubble affects the turbulence anisotropy is also related with

the duct geometry. As obtained from Fig. 7.25, the critical void fraction under which the

turbulence anisotropy reached to the constant value is larger for the smaller pipes. It

could be attributed to the strong wall effect therein.

169

Next, the bubble size effect is considered in the duct center as shown in Fig. 7.26.

Since there was only few database of the turbulence together with the bubble size, the

rough range of the bubble size was shown in Fig. 7.26. For the current database, the

bubble size DB was around 3mm in the small circular pipes, and 3~6mm in the large

ducts. It was observed from the database of the large ducts, the turbulence was more

isotropic for the smaller bubble size but more anisotropic for larger bubble size. It might

also explain why the critical void fraction under which the turbulence anisotropy

reached to the constant value is larger for the smaller pipes, which might be due to the

smaller bubble size therein.

In summary, Figs. 7.25-7.26 indicate that in the core region when the local

turbulence was dominated by the bubble-induced turbulence, under high void fraction

and larger bubble size the turbulence will be more anisotropic and the anisotropy tensor

reaches a constant value to some extent with 22.011 b and 11.033 b .

As for the bubble relative velocity and the bubble deformation effects, since there

is very few data to identify these effects on the turbulence anisotropy, in the following it

is qualitatively considered. The turbulence anisotropy in the turbulent bubbly flow could

be considered to be determined by the energy production among different direction by

the bubbles. For higher relative velocity the bubble-induced turbulence will be stronger

and more energy will be injected to the axial direction, which could make the turbulence

more anisotropic. In addition, Trygovasson’s DNSs [7.31] results showed that as

compared to the spherical bubbles, the flow with deformed bubbles behaves more

anisotropic because of the capture of the deformed bubbles in the formed bubble wake.

170


small circular pipe with DH=60mm based on Serizawa’s experimental database: under

(a) Jl =0.45m/s and (b) Jl =0.61m/s [7.2].


small circular pipe with DH=57.2mm based on Liu’s experimental database [7.7].

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b3

3

Jl=0.45m/s

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

r/R

b1

1

Jg

0.000

0.035

0.053

0.078

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b3

3

Jl=0.61m/s

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

r/R

b1

1

Jg

0.000

0.047

0.107

0.136

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b3

3

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

r/R

b1

1

Jg

0.376 0.000

0.753 0.000

0.376 0.018

0.535 0.018

0.753 0.018

(a) (b)

171


small circular pipe with DH=57.15mm based on Wang’s experimental database [7.6].

Fig. 7.23 Lateral distributions of the turbulence anisotropy tensor b11 and b33 in the large

circular pipe with DH=200mm based on Shakwat’s experimental database: under (a) Jl

=0.45m/s and (b) Jl =0.68m/s [7.13].

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b3

3

Jl=0.43m/s

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

r/R

b1

1

Jg 0.0

0.10

0.27

0.43

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

r/R

b3

3

Jl=0.45m/s

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

r/R

b1

1

Jg

0.000

0.015

0.085

0.100

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

r/R

b3

3

Jl=0.68m/s

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

r/R

b1

1

Jg

0.000

0.015

0.050

0.100

0.180(a) (b)

172

Fig. 7.24 Lateral distributions of the turbulence anisotropy tensor b11 and b33 in the large circular pipe with DH=136mm based on Sun’s

experimental database: under Jl =0.75m/s (a) along the bisector line; (b) along the diagonal line; under Jl =1.0m/s (c) along the bisector

line; (b) along the diagonal line [7.16].

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b33

Jl=0.75m/s Bisector

0 0.2 0.4 0.6 0.8 1

-0.4

-0.3

-0.2

-0.1

02x/D

H

b11

Jg

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b33

Jl=0.75m/s Diagonal

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

Jg

2x/DH

b11

0.0000.0680.090

0.1350.180

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b33

Jl=1.0m/s Bisector

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

2x/DH

b11

Jg

0.0000.0900.135

0.1800.225

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

b33

Jl=1.0m/s Diagonal

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

2x/DH

b11

Jg

0.0000.0900.135

0.1800.225

(a) (b)

(c) (d)

173

Fig. 7.25 The relation between the turbulence anisotropy tensor b11 and b33 and the local

void fraction α in the duct center for different ducts. The black and green symbols

indicate the results in the small circular pipes and the large ducts respectively.

Fig. 7.26 The relation between the turbulence anisotropy tensor b11 and b33 and the local

bubble size α in the duct center for different ducts. The black and green symbols

indicate the results in the small circular pipes and the large ducts respectively.

0

0.1

0.2

0.3

0.4

b3

3

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.4

-0.3

-0.2

-0.1

0

b1

1

Serizawa

Wang

Liu

Shawkat

Sun

b33

=-0.11

b33

=-1/30

0

0.1

0.2

0.3

0.4

b3

3

2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0

DB

mm

b1

1

Serizawa

Liu

Shakawat

Sun

b33

=-0.11

b33

=-1/30

174

7.3.2 Modeling of the anisotropic turbulent bubbly flow

In this section, we will focus on how to model the anisotropic turbulence in the

liquid phase of the turbulent bubbly flow. Before this, the physical process of the

turbulence anisotropy generation is considered for the turbulent bubbly flow as

compared to that of the single-phase turbulent flow.

7.3.2.1 Turbulence anisotropy in the single-phase turbulent flow

Firstly, transport equation of the Reynolds stress is given in Eq. (7.22) and (7.23)

ijijijij3

2DP

Dt

uuD ji (7.22)

where Pij is the production term , k

ikj

k

j

kiijx

Uuu

x

UuuP

; Dij is the diffusion

term,

k

ijk

ijx

TD

with ikjjki

k

ji

kjiijk pupux

uuuuuT

; Φij is the

redistribution term,

ijij

i

j

j

iij

x

u

x

up

3

2; εij is the dissipation rate of

jiuu , k

j

k

iij

x

u

x

u

2 and 332211 .

For the single-phase turbulent flow, the production and redistribution term plays

an important role in the turbulence anisotropy property. Models considering the slow

term (return-to-isotropy) and fast term (rapid-distortion theory) have been proposed for

the redistribution term, which is shown as follows:

ijijijkjiR PPCtuu

kC

3

2

3

22ij (7.23)

where P=P11 +P22 +P33; the first and second terms are Rotta’s model for the slow term

and isotropization of production (IP) model [7.32].

175

The turbulence anisotropy could be considered as the problem of the turbulent

energy distribution among different components. As shown in Eq. (7.21) and Fig.

7.27(a) each turbulent component is determined by the energy balance among the

production, diffusion, redistribution and dissipation. The production term due to

Reynolds shear stress and the velocity shear causes the turbulence anisotropy, while the

redistribution term due to the interaction between the pressure and strain rate of the

velocity field returns the anisotropic turbulence to isotropic and counteracts the

anisotropic production effect. Fig. 7.27(b) shows the schematic diagram of the

generation of the turbulence anisotropy of the single-phase turbulent flow in the channel.

In the near wall region, the turbulence behaves strongly anisotropic because of the wall

suppressing effect on the normal turbulent components and the existence of the high

velocity shear and Reynolds stress. While in the bulk region, the turbulence is much less

anisotropic. For the region far enough from the wall, the turbulence might be considered

as isotropic.

Fig. 7.27 (a) Contribution of each term in the energy balance to the turbulence

anisotropy; (b) Schematic diagram of the generation of the turbulence anisotropy of the

single-phase turbulent flow in channel.

176

7.3.2.2 Turbulence anisotropy for turbulent bubbly flow

As compared to the single-phase turbulent flow, the existence of the dispersed

bubbles in the turbulent bubbly flow modifies the energy balance for each turbulent

component as shown in Eq. (7.24) by introducing another mechanism of turbulence

production, diffusion, redistribution and dissipation, which affected the turbulence

anisotropy to some extent as discussed in Section 7.3.2.

2 2

3 1 3

i j S S S S B B B B

ij ij ij i j ij ij ij i j

D u uP D P D

Dt

, (7.24)

where B

ijP , B

ijD , B

ij , and B are the turbulence production, diffusion, redistribution

and dissipation terms due to bubbles.

Correspondingly, the generation mechanism of the turbulence anisotropy is

different from that in the single-phase turbulent flow. To understand the anisotropic

property of the turbulent bubbly flow and simplify the problem, the turbulence will be

considered in the area where the local turbulence is dominated by bubbles and the

shear-induced turbulence and anisotropy could be neglected such as the core region or

in the uniform flow. Therein, Eq. (7.24) could be simplified as

2 2

3 1 3

i j Slow S B B B

ij i j ij ij i j

D u uP

Dt

. (7.25)

Hereafter, the modeling of the terms related with the dispersed bubbles will be

considered.

In order to model the anisotropic turbulence property, the turbulence induced by

bubbles is considered to mimic the grid-induced turbulence as shown in Fig. 7.28,

which is often adopted to the study of the isotropic turbulence for single-phase turbulent

flow [7.32]. Therein, the turbulence was firstly induced by the wake behind the grid and

grid shear, and then return to isotropy during the turbulence decay with time. Similar to

177

that in the grid-generated turbulence, Fig. 7.29 shows the schematic diagram of

turbulence generation process with the dispersed bubbles in the core region, where the

bubbly flow was firstly simplified in the form of column like the grids. Similarly, the

turbulence is considered at the measured points as shown in Fig. 7.29. Similar to the

grids in the single-phase turbulence, there exits wake behind the bubbles and velocity

shear between the bubbles for the turbulent bubbly flow. Within the measuring time △T

in the liquid phase it might experience two periods i.e., (1) inside the bubble wake when

the turbulence Tw was produced and developed and (2) outside the bubble wake when

the turbulence decays TD and returns to isotropic. The process of the turbulence

anisotropy could be considered as the turbulence anisotropy generates in the first period

and then decays in the second period until the next bubble comes. Therefore, the final

turbulence anisotropy in the experimental measurements was strongly related with △T

and the ratio between Tw and △T, which determines the turbulence property and the

effect of return-to-isotropy. The above parameters could be considered as follows:

1cB B

B B

LlT

W W

, (7.26)

B

BB

B

wW

W

Df

W

LT

Re , (7.27)

1

ReBW f

T

T, (7.28)

where lB-B is the vertical distance between the bubbles and was obtained in Eq. 5.21; lw

is the wake length of the bubble, which is related with the bubble size DB and bubble

Reynolds number ReB.

As obtained from Eq. (7.28), Tw/△T increases with that of the void fraction. It

explains why for the turbulent bubbly flow with very low void fraction, the measured

turbulence behaves more isotropic because of the stronger effect of the

return-to-isotropy. In addition, Tw/△T increases with that of bubble size. It explains why

178

for the turbulent bubbly flow with small bubble size, the measured turbulence behaves

more isotropic because of the more isotropic turbulence generated by bubbles initially.

Therefore, during the derivation of the anisotropic turbulence model for the

turbulent bubbly flow, the above process is suggested to be considered in the turbulence

production term in the future work based on the understanding of the turbulence

anisotropy inside the bubble wake.

B B B

ij ijP A P and Re

1

B B

ij ij

fA F

. (7.29)

As for the other two terms B

ij and B , it is much more complicated and is

suggested to be considered as follows in the future model. The introduction of the

moving bubbles could be considered as the additional wall-echo effect which suppresses

the lateral turbulence and increases the turbulence anisotropy. As for the dissipation

induced by bubbles, it is suggested to be reflected in the B

ijA in the future model, i.e.,

considering the net turbulence production term due to the bubbles.

179

Fig. 7.28 Schematic diagram of grid-generated decaying homogeneous isotropic

turbulence.

Fig. 7.29 Schematic diagram of turbulence generation of the turbulent bubbly flow in

the core region where the turbulence is dominated by bubbles.

180

7.4 Summary and discussion

In this Chapter, the turbulence modulation of the turbulent bubbly flow was

studied based on the recently established experimental database including the turbulent

kinetic energy and turbulence anisotropy. The following findings were obtained:

(1) In order to quantify the solely bubble-induced turbulence in a wider range

of void fraction, the turbulence was investigated in the duct center where it

was dominated by the bubbles. It was found that the relation between the

turbulence and the void fraction was affected by the duct geometry.

Comparing to the small circular pipes the stronger turbulence could be

induced in the large ducts. In addition, the turbulence in the small circular

pipe was found to be closer to prediction of the potential theory. In the large

ducts, the turbulence model needs to be modified to get better prediction.

(2) The present study on the interaction between bubble-induced turbulence and

wall-induced turbulence indicates that comparing to the core region, the

bubble-induced turbulence under the same void fraction was reduced by the

wall effect, which made the turbulence more homogeneous in the turbulent

bubbly flow. In addition, the turbulence reduction due to existence of the

bubbles was discussed to consider the bubble-effect on wall-induced

turbulence. The acceleration laminarization and suppression of the ―bursting

events‖ of the coherent structures in the near wall region were proposed as

the mechanisms responsible for the turbulence reduction therein.

(3) The present study on the turbulence anisotropy indicates that the existence

of the bubbles could affect the turbulence anisotropy such as causing the

turbulence more anisotropic in the duct core region but more isotropic in the

near wall region. How the turbulence anisotropy is affected by bubbles is

related with the void fraction, duct geometry, and bubble size. Under higher

void fraction or larger bubble size, the turbulence anisotropy tensor was

found to be more anisotropic and reach a constant value to some extent.

181

It is notable that since the database for the liquid phase turbulence is very

limited so far, the present analysis was only carried out for several databases. Therefore,

it still needs further validation of the above observations based on new turbulence

database for the turbulent bubbly flow especially in the large ducts to consider the solely

bubble-induced turbulence in the future. In addition, for the anisotropic turbulent model,

more databases about the bubble and bubble-induced wake are also required.

182

Reference

[7.1] Hosokawa, S., Tomiyama, A., 2010, Effect of bubbles on turbulent flows in

vertical channels, 7th International Conference on Multiphase flow. FL USA.







[7.5] Michiyoshi, I., Serizawa, A., 1986, Turbulence in two-phase bubble flow, Nucl.

Eng. Des. 95, 253-267



Flow 13, 327–343.

[7.7] Liu, T.J., Bankoff, S.G., 1993, Structure of air-water bubbly flow in a vertical


Transfer 36, 1049–1060.

[7.8] Serizawa, A., Kataoka, I., 1990, Turbulence suppression in bubbly two-phase

flow, Nucl. Eng. Des. 122, 1–16.

[7.9] Lance, M. , Bataille, J., 1991, Turbulence in the liquid phase of a uniform

bubbly air-water flow, J. Fluid Mech., 222, 95-118.

[7.10] Lance, M., Bataille, J., 1991, Homogeneous turbulence in bubbly flows, J.

Fluids, Eng., 113, 295-300.

[7.11] Wijngaarden, van L., 1998, On pseudo turbulence, Theoret. Comput. Fluid

Dynamics 10, 449-458.

[7.12] Shawakat, M. E., Ching, C. Y., Shoukri, M., 2007, On the liquid turbulence

energy spectra in two-phase bubbly flow in a large diameter vertical pipe, Int. J.


183





Part 2: velocity fluctuations, J. Fluid Mech. 466, 53-84.

[7.15] Hibiki, T., Ishii, M., Z. Xiao, 2001, Axial interfacial area transport of vertical

bubbly flows, Int. J. Heat Mass Transfer 44, 1869-1888.







[7.19] Politano M. S., Carrica P. M., Converti J., 2003, A model for turbulent

polydispers two-phase flow in vertical channels, Int. J. Multiphase Flow 19,

1153-1182.

[7.20] Hosokawa, S., Tomiyama, A., 2009, Multi-fluid simulation of turbulent bubbly

pipe flows, Chem. Eng. Sci. 64, 5308-5318.

[7.21] Rzehak, R., Krepper, E., 2013, Bubble-induced turbulence: Comparison of CFD

models, Nucl. Eng. Des. 258, 57-65.

[7.22] Yokomine, T., Shimizu, A., 1995, Prediction of turbulence modulation by using

k-ε model for gas-solid flows, Advances in Multiphase Flow 191-199.





[7.25] Inoue, A., Koga, T., Yaegashi, H., 1976, Void fraction, bubble and liquid

velocity profiles in a vertical pipe, Trans. Japan Soc. Mech. Eng. 42, 2521-2531.

[7.26] Sullivan, J. P., Houze, R. N., Buenger, D. E., Theofanous, T. J., 1978,

Turbulence in two-phase flows, Proc. 2nd

CSNI Specialist Meeting, Paris,


184

583-608

[7.27] Ohba, K., Yuhara, T., 1982, Study on vertical bubble flow using laser Doppler

measurement, Trans. Japan Soc. Mech. Eng. 48, 78-85

[7.28] Aoki, S., 1982, Eddy diffusivity of momentum in bubbly flow, M.S. Thesis,

Kyoto University.

[7.29] Lee, S. G., 1986, Turbulence modeling in bubbly two-phase flow, PhD Thesis,

RPI

[7.30] Bolotnov, Igor A., 2013, Influence of bubbles on the turbulence anisotropy, J.

Fluids Eng. 135, 051301.


properties of bubbly flows, J. Fluid Mech. 495, 77-118.


185

Chapter 8

Conclusions

8.1 Summary of dissertation

Io order to improve the understanding and modeling of the turbulent bubbly

flow especially in the large square duct, the present dissertation contains three parts.

Firstly, the state-of-the-art of the upward turbulent bubbly flow was reviewed in Chapter

1 and the existing database was collected in Chapter 2. In second part, the numerical

evaluation of the existing models was carried out based on the experimental databases

in the large square duct. The third part was contributed to improving the understanding

of the flow characteristics and the physical processes therein including the flow

developing processes, the axial velocities of the liquid phase, the bubble deformation

and the turbulence modulation in the liquid phase. The main achievements in each part

could be summarized as follows:

1) Numerical evaluation (Chap. 3)

It indicates the existed model might not be suitable for the turbulent bubbly flow

in the large noncircular ducts because of the different physical processes therein as

compared to that in the small circular pipes. The detailed analyses of the physical

process of the turbulent bubbly flow in the large noncircular ducts are suggested to

improve the model therein.

2) Flow developing process (Chap. 4)

During the developing of turbulent bubbly flow, there exist at least two stages

which are forming the bubble layer and the velocity developing based on the formed

bubble layer. During the bubble layer formation at the first stage, the length required to

186

form the bubble layer was considered and found to be less than that required to develop

the liquid phase boundary layer with the similar thickness. At the second stage, the

length required to lift up the liquid phase velocity near the wall was analyzed, and the

criterion to judge the flow developing status was suggested.

3) Axial velocity in the liquid phase (Chap. 5)

For the turbulent bubbly flow in the large square duct, the formation of the

typical M-shaped velocity profile was noticed and studied by considering the index of

flow type, the formation conditions, the peak locations of the velocity profile and

velocity characteristics. Therein, two layers with the different flow types were observed

and analyzed between the wall and the duct center, which are the pressure driven and

shear driven flows. The formation of the M-shaped velocity profile was found to be

determined the duct geometry which dominates the lateral void fraction distribution and

the wall drag resistance. Based on the flow characteristics of the turbulent bubbly flow

in the large square duct, two-layer model and analysis was suggested to improve the

understanding and modeling therein.

4) Bubble deformation (Chap. 6)

The void fraction effect on the bubble deformation for the turbulent bubbly flow

was found which showed that the bubbles behaved more prolate with the increase of the

void fraction as compared to the single-bubble case. A new correlation for the statistical

bubble deformation was proposed and validated combining the bubble size and void

fraction to consider the void fraction effect. Numerical validation of the void fraction

showed that only for the cases with better developed status, the predictions of the void

fractions were in well consistent with the experimental measurements. It indicates the

importance of knowing developing status to model the turbulent bubbly flow.

5) Turbulence modulation in the liquid phase (Chap. 7)

The quantification of the solely bubble-induced turbulence in a wider range of

void fraction was carried out by focusing the turbulence in the duct center. The relation

187

between the turbulence and the void fraction was found to be affected by the duct

geometry. As compared to the small circular pipes the stronger turbulence could be

induced in the large ducts. The interaction between bubble-induced turbulence and

wall-induced turbulence was considered by comparing the turbulence in the core region

and near wall region. It indicates that as compared to the core region, the

bubble-induced turbulence could be reduced by the wall effect. The mechanisms of the

turbulence reduction of the turbulent bubbly flow were proposed which are the flow

acceleration laminarization and suppression of the coherent structures in the near wall

region. The investigation of the turbulence anisotropy indicates that the introduction of

the bubbles could affect the turbulence anisotropy such as causing the turbulence more

anisotropic in the duct core region but more isotropic in the near wall region. However,

how the turbulence anisotropy is affected by bubbles was found to be related with the

void fraction, duct geometry, and bubble size.

8.2 Recommendation for future work

1) To understand the flow developing process

There are several interesting observations during the developing of the turbulent

bubbly flow such as two-stage developing and the almost constant thickness of the

bubble layer δB. In addition, the understanding of the developing status is also important

for the flow modeling. However, the current database is still limited to understand the

detailed developing process of the turbulent bubbly flow. Therefore, the boundary layer

flow type is suggested for the bubbly flow.

2) To understand the secondary flow

Currently, the secondary flow effect in the turbulent bubbly flow was neglected

by considering the experimental data of the lateral turbulent normal stresses. So far

there is no direct experimental information about it, which is suggested in the future

work.

188

3) To understand the bubble deformation

Currently, the void fraction effect on the bubble deformation was considered for

the mean bubble size around 3.5mm~5.5mm and limited void fraction range. For the

wider range of the bubble size and flow conditions, it still needs further study in the

future.

4) To understand the turbulence modulation

The mechanisms here proposed for the turbulence reduction of the turbulent

bubbly flow still needs further experimental measurements including the velocity and

turbulence in the near wall region to confirm.

189

Nomenclature

Latin

ai Interfacial area concentration

A Dimensionless parameter to show flow developing status

AS Anisotropy matrix related with shear-induced turbulence

AB Anisotropy matrix related with bubble-induced turbulence

b Turbulence anisotropy tensor

Cε Model constants in η-ε model of the turbulent bubbly flow

Cε1 Model constants in η-ε model of the turbulent bubbly flow

Cε2 Model constants in η-ε model of the turbulent bubbly flow

CεB Model constants in η-ε model of the turbulent bubbly flow

Cμl Model constants related with shear-induced eddy viscosity

Cμb Model constants related with bubble-induced eddy viscosity

CL Lift force coefficient

CT Turbulent dispersion force coefficient

CVM Virtual mass force coefficient

DB Bubble size

DH Duct hydraulic radius

DHB Maximum horizontal bubble dimension

D Diffusion term Reynolds stress transportation

Dmax Maximum bubble size

DSM Sauter mean diameter

DV Axial bubble dimension

E Bubble aspect ratio or bubble deformation

Eα Bubble aspect ratio for bubbly flow

190

ES Bubble aspect ratio obtained by single-bubble correlation

Eo Eötvös number based on DSM

EoH Eötvös number based on DHB

fb Bubble frequency

fl Friction factor

F Local net driving force

g Gradational acceleration vector

Jk Superficial velocity of phase-k

k Turbulent kinetic energy

kBa Asymptotic value of bubble-induced turbulent kinetic energy

le Characteristic length of the most energetic eddy

ls Mixing length of shear-induced turbulence

lb Mixing length of bubble-induced turbulence

lB-B Vertical bubble-bubble interval

lw Length of the bubble wake

LB Length from the inlet to form bubble layer δB

Lc Chord length

Lδl Length from the inlet to form liquid-phase boundary layer δl

Lp Length required

M Morton number

Mik Interfacial force exerted on phase-k

Mikd

Interfacial drag force exerted on phase-k

MikL

Lift force exerted on phase-k

Miknd

Interfacial non-drag force exerted on phase-k

MikP

Interfacial term due to pressure exerted on phase-k

MikS

Interfacial term due to shear stress exerted on phase-k

MikT

Turbulent dispersion force exerted on phase-k

191

MikV

Virtual mass force exerted on phase-k

MikW

Wall force exerted on phase-k

Mwk Wall shear force exerted on phase-k

pk Pressure field in phase-k

P Production tensor in Reynolds stress transportation

P P=trace (P)/2

Re Reynolds number

Si Interfacial tensor in turbulent kinetic energy transportation

Si Si=trace (S)/2

SB Surface area of the bubble

te Characteristic time of the eddy

tk Turbulent kinetic energy in Chapter 7

TD Time period of the bubble wake decay

Tw Time period of the bubble wake

We Weber number

vk Fluctuating velocity vector in phase-k

k kv v Reynolds stress tensor in phase-k

kV Phase averaged velocity vector in phase-k

Vk Instantaneous velocity vector in phase-k: Vk= kV + vk

VT Terminal velocity

VR Relative velocity between bubble and liquid phase

VB Volume of the bubble

Wk Axial velocity of phase-k

Wlm Maximum of Axial liquid-phase velocity

W*

Dimensionless velocity

x Coordinate component, horizontal direction

X*

Dimensionless location

192

xp Location of the liquid-phase velocity peak

y Coordinate component, horizontal direction

z Coordinate component, flow direction

Greek

αk Void fraction of phase-k

Mass quality

δl Thickness of the liquid phase boundary layer

δB Thickness of the bubble layer

ε Dissipation rate of the turbulent kinetic energy

ε Dissipation term Reynolds stress transportation

Φ Redistribution term Reynolds stress transportation

γ Ratio between maximum void fraction and that in duct center

μk Dynamic viscosity of phase-k

υb Eddy viscosity due to bubbles

υs Eddy viscosity due to shear-induced turbulence

ζ Surface tension between liquid and gas phase

ζε Model constants in k-ε model

ρk Density of phase-k

τB Bubble-induced Reynolds stress tensor in liquid phase

τk Shear stress tensor in phase-k

τkT

Turbulent Reynolds stress tensor in phase-k

Superscript

B Bubbly flow

c In the duct center

S Single-phase flow

193

w Near the wall

Subscript

0 Single-bubble cases

α Bubbly flow cases

b or B Bubble-induced

c Duct center

i Interface-induced

in Insider the bubble layer

k Phase number

g Gas phase

l or L Liquid phase

m Mean value

out Outside the bubble layer

s or S Shear-induced

t Two-phase flow

w or W Wall-induced

duct center

S Single-phase flow

w Near the wall

Subscript

0 Single-bubble cases

α Bubbly flow cases

b or B Bubble-induced

194

c Duct center

i Interface-induced

in Insider the bubble layer

k Phase number

g Gas phase

l or L Liquid phase

m Mean value

out Outside the bubble layer

s or S Shear-induced

t Two-phase flow

w or W Wall-induced

Documents

Title Study on Upward Turbulent Bubbly Flow in Ducts ...repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/...express how grateful I am to my father Mr. Liu-Xian ZHANG, my mother